Is it possible to give your Unseen Servant the ability to fly for the duration of that spell?
I'd say no. An unseen servant is not a creature, so can't be targeted by the spell.
you could cast mud buddy, which is basically the same as unseen servant and then cast fly on it. If it needs to be invisible then you could also slap invisibility on it.
It's not ideal, but since unseen servant can't be directly targeted by harmful spells, I don't see any way to target it with beneficial spells like fly.
Thank you both for the answers and suggestion.