The NX-02 could probably take up the NX-01's course before they headed into the Delphic Expanse. So you have the opportunity to run into other species. There are lots of species to meet who haven't been cast in that stronger than tritanium material known as canon. You can always use species from other sources. Being an old Starfleet Battles player, I like to use some of the species from there. You have Hydrans, Lyrans, Kzinti, the Orions. You can run into other Trek species and begin the formal first contacts with them. Try meeting the Tellarites "officially" and see how they earned their reputation of arguing for the sake of argument.
Throw in run ins with Klingons every now and then, but not too often. Don't be too afraid to go against continuity or canon, unless you know your players will burn you at the stake for it.
Sorry, I just know that some people are just a little touchy about it.
Don't forget the old standby's of epidemics on colonies that need urgent supplies, troublesome passengers, and god-like aliens.
Derelict ships and space stations are also fun places to visit. Just watch a few episodes of TOS or Enterprise to get a couple of ideas.
"Retreat?! Hell, we just got here!", annonymous American Marine, WWI
"Gravity is a harsh mistress....", The Tick