I'm assuming for this answer that the person you're looking for is indeed one of your ancestors and not someone on a collateral line.
One difficulty with doing this kind of search is simply keeping track of where you have already looked. Just to clarify things, you could try filling out a series of pedigree charts or ahnentafel (ancestor tables) of your own so you can see your data in a different way.
The website Genealogy Junkie has templates for a compressed 7-Generation Pedigree Chart where generations 1-3 run sideways to save space; it includes Ahnentaefel numbers. Filling out one of those charts for yourself and your close-by ancestors will give you an example of how the numbers work.
The advantage of using the ahnentaefel in a spreadsheet instead is that you can put the data in columns and the spreadsheet can be extended indefinitely.
However you want to keep track of your data, make a list of all the people that are at the farthest reach of the FamilySearch FamilyTree relationship calculator (the 15-generations out which you mentioned in your question).
Then, go through those people one by one in family tree, and use the the Fan Chart to display all the known ancestors for each person. You can do this by selecting Fan Chart from the drop-down menu at the upper left of the screen.
This will allow you to see several generations of that person's ancestors at once, making it easier for you to spot the person you're looking for than it would be if you traced each line separately.
Once you find people of interest, use the Watch feature on their profile (see the star) to keep track of them, and add them to your notes and ahnentafel so you can find them again.