Can I put drip edge underneath felt without removing roof?
It's virtually impossible to get drip edge under the bottom edge of a roof without redoing the whole roof.
You might be able to improve it by using flashing tape to seal the drip edge to the felt, if there's enough flex in the (shingles, I assume - you didn't say) on top of it to get that in without breaking the (shingles?)
You might also be able to slip a strip of felt (non-sticky) up under the lowest shingles on top of the joint, which won't require as much flexing if they are stiff, and can probably be shoved further up than the sticky flashing tape can be adhered. You'll want a few dabs of tar or other compatible roofing adhesive to keep that in place once it's shoved in as far as it will go.