Well, there might be a few possibilities.

So, I am in the process of planning a Star Wars story, and the idea I have is that a Jedi Padawan could have escaped Order 66 and now wanders the galaxy so as to remain on the move and not be as easily detected. One thing I thought that would be great for character development is that he could (without realizing it) pass the Trials, and I have planned realistic scenarios for each Trial. I thought at some point he could become a Knight, but here's my problem... I have no idea how he could be promoted. He'd have a travelling companion at some point, but he or she would not be Force-sensitive, and the OC's Master will have died long ago. As I haven't officially watched Rebels yet (but I know quite a bit about it), I don't want him mixing up with any canon characters from there. So technically, no other Jedi will have witnessed his Trials to say he is worthy of Knighthood. And I think we can pretty much all agree that it would be selfish of him to dub himself a Knight, or demand that another Jedi knight him. So with all of this, would he remain a Padawan forever? I'm not against that idea, by the way; but I know that option is a possibility in this scenario.

The ones I can think of being:

1) A Jedi Temple.
While the rules on Force ghosts aren't quite clear (in either canon, so far as I know) it might be possible for your character to stumble across some lesser known Jedi outpost and be confronted by a guardian or spirit there, and be promoted by them.

2) Someone who knew Jedi tradition well enough to decide if your character would be worthy.
Perhaps a historian of some description, a member of another Force using group (depending on whether this is Disney or Legends), a notable associate of a Jedi Master, or someone who worked in the Temple itself. I know you said it would be selfish of him to Knight himself, but perhaps it would be different if he didn't make that judgment himself? I dunno.

3) A Jedi Holocron.
While the Holocron gatekeeper (provided those are still a thing in Disney canon) wouldn't have the authority to actually Knight anyone, it would certainly know better than most when a person is worthy of the title, and after a while of being considered a Knight in all but name, why not just accept it? Goes back to the Knighting himself thing, but I'd say it would be entirely reasonable to consider yourself a Knight if the closest thing to a Jedi Master you're likely to find says you've earned it. Suppose it depends on the character, though. And there's the rather significant problem of how to find a Holocron.

4) A former Jedi.
While Jedi Masters leaving the Order wasn't a common thing, there are enough instances in Legends to justify making one of your own, and if it's Legends, then there's a whole new canon to work with. Of course, that also means there's a whole new canon to trip over, but I'm sure you could make it work. As long as you could find a plausible reason for the Master to leave the Order (the horrors of the Clone Wars, for instance) then I see no reason why they couldn't survive Order 66 long enough to meet and promote your character. Hard to track a Jedi who's not a Jedi anymore, after all.

Hope that's a least a little bit helpful, and if you need any more help feel free to drop me a line. I may not be quite the expert I used to be, but Star Wars remains my all time favorite fandom.