In 2 Religious Clan quests I wouldn't get the reward if I didn't exit the room and enter again. It would be nice if we could start a new quest right away after finishing the current one, without exiting the room and entering again, which is annoying.

Yes. This would be great. I also wish the NPCs said more when you finished. Like..

Great job getting the canned food. I got another task that needs done. You interested?

[Tell him you are]   [Tell him your busy]

I've found that clicking on the character tab, and then clicking on the location tab again, will solve this problem. I don't have to exit the room anymore.