As others have discussed, you can try for a Mysterious Stranger while you have the merc you want in your party. It’s very slow now, but you’ll find one eventually. If the merc you want is a useful member of a good team, it’s easy enough to just play that team and check the mystery boon in every bounty just in case. Ideally with an alternate path you can take if the “boon” is the Spud ME. 
Another method I used shortly after the current task system was implemented, when I still had a lot of mercs needing tasks, was to do tasks for the random mercs on my task board for a few bounties, then when I stop playing for the day (often when the least desirable mercs reached a slower task, like 7, 13, or 15, or one of the oddball ultra-grindy tasks some mercs have) I’d abandon one or more mercs to cycle the task board in the hope of getting a more useful merc onto it the next day. Repeat as needed until you get all the important mercs done and/or the randos progress far enough that you may as well finish off their tasks to remove them from the pool.