So far as I know, there is no medical institution offering tests to random people who think they had it, just controlled investigations of special populations (prisons, nursing homes, etc.).
I was looking for some more possible answers to Ray’s question, and discovered that WebMD is following the now-common practice of buttering both sides of their bread.
https://blogs.webmd.com/webmd-doctors/20200415/wondering-if-you-could-have-had-covid19-last-fall

This post was updated on 4/27/20 to reflect new information.

As public health experts try to stop the spread of the virus that causes COVID-19, doctors are also working to curb the spread of misinformation.

One story that has recently spread rapidly suggests that many Americans may already have been exposed to COVID-19 in November and December of 2019. Many people are sharing stories on social media of unexplained fevers, or flu-like illness during those months and wondering if they may have had COVID-19 without realizing it.
…
Though it’s tempting to believe these stories and hope for a quick end to lockdowns and “stay at home” orders, this type of misinformation can be dangerous. It can undermine the policies that have helped to make California a success story, such as instituting social distancing directives earlier than other places in the U.S.

When a reporter from Slate asked Stanford researchers to weigh in on this story, they explained that their research does not suggest that COVID-19 was in this country in the late fall or winter of 2019.

The science of genetics also provides information about the global spread of COVID-19 and allows scientists to track its path across the world. Based on minor mutations that subtly change the genetic code of the virus, researchers have traced the origin to China during the November and December time frame. …
However, there is now new evidence that the virus may have been circulating in the U.S. as early as January and February. There may have been deaths related to the virus that have not been documented because we were not testing for COVID-19 during this time. There is also some evidence that many more people have been infected than previously thought. What all of this may mean is still not clear. …
The biggest takeaway from these findings is that researchers are following every crumb to identify the origins and path of the spread of COVID-19 and trying to share this information as quickly as possible to keep people safe.

Currently, researchers are working furiously to identify people who were previously exposed and possibly immune throughout the country. These findings will inform policies that can help us ease up and open the economy in a safe and effective manner. It is unlikely that we will safely achieve herd immunity without an effective vaccine.

To summarize: we know that the virus was in China in November & December; we’ve identified slightly different strains in the USA in January & February; but, we weren’t testing everybody that early, and more people had the coronavirus than we once thought, and we really want to find out how many have already had it, BUT DON’T YOU DARE SUGGEST that YOU were infected last fall — that’s unscientific anecdotes, and only WE have the data.

You aren’t one of the crumbs of evidence we want to pursue.

Questions, seriously: How are the researchers dating the advent of the strains they found in January & February? How long were those in the country (earliest known date of infection)? Where did the infected people get those particular strains, and where were/are the people who had the genetic progenitors?

This opinion piece is a bit more welcoming of the idea of early infection, but only back to December. It gives us the state of the art as of April 7, so is somewhat out of date, but is pushing the same line (in the last sentence) as was WebMD 20 days later.

https://www.health.com/condition/infectious-diseases/coronavirus/could-i-have-already-had-coronavirus

Scientists believe the virus emerged in China in December 2019. International travel continued throughout January, as cases popped up in other destinations—such as Australia, Bangkok, South Korea, France, the UK, Japan, and Russia. The US confirmed its first COVID-19 case on January 21. But the virus was already spreading other places, too.

By the time we recognized the start of the pandemic in the US, it’s very possible it was already here and well-established.
…
It’s not out of the realm of possibility that this virus has been spreading in the United States since the very early part of the year, perhaps in early January or very late December, Jagdish Khubchandani, PhD, MPH, an associate chair and professor of health science at Ball State University in Indiana, tells Health. “Given the timelines and incubation period reported by studies, it could be possible that many Americans had this infection way back and we are just now getting to see the severe most cases,” he says.
…
Dr. Schaffner senses that a lot of people will want to know if they’re immune. “We’d like to test first responders; many people of a certain age will want to know,” he says. Dr. Schaffner adds that it’s about making a test streamlined, so it can be made widely available.
…
In the meantime, even if you think you may have been infected, it’s important to act like you’re supremely vulnerable—for your sake, and for others’ sakes—because there’s no way to know for sure.

So, maybe Ray and his wife just had some weird variant of the flu, but the researchers at this point seem so convinced they already know where to look for the answers, they aren’t really interested in finding out if they actually do know all the answers.

And You-Book/Face-Tube will suppress anyone who gets different answers anyway.

I have taken the flu vaccine every year since it has been offered because I worked (as a vendor) in many different health care facilities. I have not had the flu in over ten years. However, right around Thanksgiving, I started feeling signs of an oncoming upper respiratory infection. By the following weekend – the first weekend in December – I was well and truly sick. My wife got it about the same time I did, but hers was worse.

I was hard down for about three days. I got better fairly quickly but it was about another two weeks before I felt normal again. My wife was hard down for almost two weeks. After five days, I got worried and took her to the local immediate care center. They gave her a flu kit and an inhaler because she could barely breath – which was one of the symptoms we both had. The inhaler did the trick and she immediately started feeling better. But it was the end of January before she felt normal again.

Of course, at the time nobody had even heard of CoVid-19, or Kung-Flu, or whatever you want to call it, so we thought it must have been some kind of weird flu. Some of our symptoms were the same as a classic flu, but some were not. Now, when I read the symptoms of CoVid-19, they match exactly the symptoms that we both had.

Now, I understand that I could be mistaken. We both could have had some weird variant of the flu. However, I think it would be worth my while to get the antibody test to find out. But how does someone who is now asymptomatic go about doing that?

Wow, that’s astounding. EARLY December.

https://newsthud.com/breaking-supreme-court-says-justice-ginsburg-hospitalized-with-an-infection/

https://preview.redd.it/qk4t4vt4wdu41.png?width=640&crop=smart&auto=webp&s=9474baa65f14b13e5a6232484272abb9e31f8c27

I know they aren’t required to test for the flu. But here (NYS), during flu season, it seems to be SOP unless they’ve run out of tests.

But if many don’t, because they aren’t required to do so, that only makes the possibility, to my mind, of Wuhan Pox being around much sooner more plausible.

It would seem to me that a national Center for Disease Control and Prevention would be interested in having stats on how many people have flu or virus infections as a matter of course, whether fatal cases or not.
Maybe now they will start mandating tests on everyone for everything all the time.