One of the characters in my new game is going to be a Twi'lek Bounty Hunter/Survivalist with an Obligation of Under Contract (this is designed for a Smuggler but I allowed the player to use it). The details of the Obligation is that the PC has taken a job from the Empire to hunt down an NPC, but is reluctant to do it but the bounty was too good to refuse.

I'm not sure how I can work this to have story elements that work with increasing or decreasing the Obligation of the PC. Clearly finding the NPC and eliminating it would probably close the case and remove the Obligation, but this seems quite final. Does anyone have any ideas?

Have him contacted by his handler or an imperial accompanied by Stormtroopers that in lieu of that outstanding contract he has yet to fulfil they have another bounty for him to deliver to them.

As long as his original contract remains unfulfilled he is liable to deal with other tasks until he completes his original contract this way it is a genuine drawback that it's supposed to be.

Have you watched Killjoys?

This sounds alot like that.

Better yet...have the local Imerial Garrison pick him/her up as a person of interest and bring her to the local Governor's office which the person that they signed the contract with is waiting to "remind" them of the terms of their deal but they need to keep watch on them while they build a case to take out his network.

They then get released...weather they get a Imperial "Tune-up" or not is upto you and the PC

Better yet...have the local Imerial Garrison pick him/her up as a person of interest and bring her to the local Governor's office which the person that they signed the contract with is waiting to "remind" them of the terms of their deal but they need to keep watch on them while they build a case to take out his network.

They then get released...weather they get a Imperial "Tune-up" or not is upto you and the PC

Does it make me a terrible human being for thinking that would make a great start to a "Usual Suspects" style session? Was your bounty hunter ever part of a barbershop quartet?

Some other suggestions:

The Empire loves to micromanage! Have a governor or moff get all up in his bidness.

The Empire loves to ... "renegotiate" on the fly! Oh, and while you are doing this, we're commandeering your ship/making you train this stormtrooper reject/"straightening out" the governor's kid with his silly spice addiction... the possibilities are endless!

Not only that, but if word gets out about his Imperial entanglements, there are probably a lot of places where the criminal element may take offense to that. After getting that Imperial "tune-up" he just wants to have a nice quiet drink when some loudmouth starts talking about how that guy over there works for our "evil overlords" and smuggler fisticuffs ensue!

Possibly they placed the bounty on the individual in question because he or she has something incredibly valuable to the Empire. Your PC could "solve" the obligation and turn in the NPC for the bounty only to discover that the issuing authority is now very unhappy, incredibly suspicious of the bounty hunter, and demanding that the job isn't complete until the McGuffin is located and returned.

They might even be willing to give 10% of the bounty for doing part of the job, implying that continued failure would be lethal unfortunate.

I'd be more worried if they're keeping tabs on the character and their friends... after all you never know how useful they'll be in infiltrating the Rebel Alliance after all?!