Historically, people rarely emigrated alone. Life was too precarious and difficult to handle without the support of a family network and this is especially true for populations undergoing group displacement, like the Loyalists who resettled in Canada after the US War of Independence. If you have a brick-wall ancestor, read up a little on the community they lived in and ask yourself, how likely is it that none of their relatives lived nearby?
For most North American cases, it is not likely your ancestor was a true "stray," unconnected to everyone else. In the past, many genealogists were not particularly focused on identifying collateral relatives because they were trying to get a generation further back, records tend to run out at specific points in time, and they assumed this stranded everyone in that generation without clues (not in itself a great strategy, as land records and the death registrations of the long-lived often prove).
However, the advent of DNA testing changes the potential return-on-investment to be had from spending time identifying your brick-walled ancestor's siblings.
Firstly, if you know your ancestor's surname, it takes only a few minutes searching FamilySearch tree/records, FindAGrave, and Ancestry to locate people in the community of interest with that surname, born at the right time. If the name is rare the list will be short. Short lists and rare names are always easier to work with, so if your brick wall has that attribute, don't waste the opportunity to exploit it.
Identify the spouses of these potential siblings and check all their vital records and attached sources to see if other researchers have identified legitimate clues about the origins of these people. Then, add the potential siblings, their spouses, and their children to your DNA-test-connected family tree. Assuming your brick-wall is within about seven, maybe eight generations (4th/5th-great grandparents), if at least some of these newly-found siblings are in fact your biological relatives, some of their descendant testers (if they have descendant testers) should show up in your DNA matches. If you are using Ancestry, you would find new ThruLines suggestions in support of any correct theory after waiting a day or two for the tree additions to recalibrate with your match list (with the caveat that you need to check for additional relationships explaining the DNA shared).
Now, I'm writing this blog because last week someone submitted an incorrect FindAGrave memorial edit (thinking they had found an ancestor's burial, but cemetery documentation indicated it was an infant with the same name). Looking into the situation, I discovered a cluster of people with a moderately rare surname in the community who had been born from 1797 to 1811 (and in a consistent sequence of one born every year or two). They were all American and by checking my local genealogical history books for the area, I found three of them were closely associated (although not explicitly identified as family) in a chapter documenting one of their spouses. These three, plus a couple of others, had considerable name overlap showing in their children, as well as potential naming of children in honor of their married-in aunts and uncles. Looking at the documentation across the entire group, there were three vital registrations suggesting an origin in Dutchess County, NY. No online tree considered these people as siblings.
In fact, a couple of them have been assigned to family origin groups that have nothing (else) to do with this particular community and there is a notable absence of name repetition across the generations in their current placements. None of these assigned family relationships have an evidentiary source to prove the connection (and none are from Dutchess NY) — in all cases researchers have simply selected a surviving American baptism record from roughly the right year of birth and decided it was for the person in question who ended up in Canada.
Having developed this tantalizing theory of relatedness, there is nothing else I can do because these are not my biological relatives. However, if descendants knew of this theory and have DNA tested, they could check its merits within 72 hours, just by following the steps above. If the theory of siblingship is supported by DNA, then they would have a specific place to go to for further records checks (Dutchess) and a list of significant personal (and potentially maiden surnames) to be on the look out for. They'd also have other unexplained DNA-group matches to potentially help find further back generations. This is a much better investment of time and resources, as compared to the ole' throw-a-dart-at-the-coincidentally-similar-baptism-record-and-pray method of the pre-DNA testing era.