I don't think your revision means the same thing as what's being conveyed in the original example, though the general gist is the same.
I had given up once before, but I felt that there might still be a possibility, so I decided to give it another try.
However, the original example reads more like this to me:
I had given up once before, but I felt like there must still be a possibility, so I decided to give it another try.
Substituting the 気がした for 気がしてならなかった in this case gives off the implication that the belief towards assurance that there is a way is leaning on the side of believing in the possibility rather than simply stating that there might just be one, much like in English you'd say "I think it's possible" versus "It must be possible."
I might not be explaining in the best way possible, but that's my understanding of it.