1 - Make it so hits last for 3 months instead of 1. It would encourage more investigation since people would have the time to do it, one of my characters used to help assassins by doing this. They would ask an assassin who had hits on them, and then go and try to locate them and build a dossier on them. As well, if a hit lasted 3 months, it'd be harder for people to wait it out. One month isn't long, it's fairly easy to wait out right now. Especially if it's an expensive bounty. Knowing that it'll go away in a month makes it an easy choice to not remove it or worry about it.
2 - Make it so the book in the guild hall actually listed people who were assassinated. Right now it feels like money is being thrown into the void, without knowing if there are even any active assassins who will pursue the hit. Seeing if the hit is complete would make the system more satisfying, knowing that gold was spent well and not just thrown out. It would also let people know if a hit needs to be refreshed or not once the time period is up.
2b - Include how much the hit was for, so people can see the numbers people were willing to put out. 1 million gold hits are exciting! Aim for the high score!
The next two ideas are things to try and increase the value of an assassin hit, to make it worthwhile to put a hit on someone other than a settlement leader:
3 - Make it so an assassin hit on any settlement official would remove them from their position and make it so they can't be re-hired for 1 irl month. Things like magistrates, trade ministers, anything listed as a role when you speak to the settlement npc. The downside to this is that cliques might use it to entirely destroy anyone who takes over a settlement, I'm not sure if there's a way to prevent that kind of abuse. But on the other hand, it could also encourage turnover if those positions are filled by new people.
4 - Make it so if someone is assassinated, they can't return to life for 48 hours. Their shop and house timer will not go down during this time, so there's no risk of losing property. The value would be to remove someone from the narrative for a few days. It would add weight to dying, and would be something only assassins could do. The class is application locked and has expectations, which limits people from being able to do a lot of damage with it.
Personally I like 4 more than 3. I think it has much lower grief potential for ruining people having fun, and could possibly be used to the same end since it removes the character for a few days.