That’s the original implication, but it doesn’t make much sense either way.
If he was thrown from the crash (didn’t eject), then he probably couldn’t have flown far without suffering major injuries or death on impact. In that case, Finn or more importantly the First Order would have found him. And if both Finn and Poe were lying unconscious in broad daylight fairly near to the TIE, then the First Order would have been completely blind to have missed them in their search, especially knowing exactly who they were looking for and where. I assume in all of this that the First Order has already searched the TIE by the time Finn awakens, since if they arrived after that they would have quickly tracked him down anyway.
So in an attempt to make the First Order something other than completely incompetent, I thought that his statement ‘I woke up at night, no you, no ship, nothing’ could refer to him being without Finn or a working ship and having to steal the Stormtrooper squad’s transportation. It’s a stretch, but it is no less illogical than the other implication.
Perhaps if I add the sound of two TIE fighters taking off (perhaps gunfire), and this is what awakens Finn…