What area are you camping at? If you are down in sherwin creek area then laurel lakes trail head is just south a few miles. The views up there are straight out of an Ansel Adams photo shoot. You may want to bring some fishing gear with you on this one for the lakes. Actually if you like to fish, you want to carry your gear with you full time up there!!
If your up around twin lakes or lake mary then you may want to hit the inyo craters area off the scenic loop drive which takes you up towards deadmans creek where there are miles of easy 4low volcanic sand with scenic areas and creek crossings in and out of the trees. You can figure out different loops which can take you back to the scenic road or all the way back down into the town of mammoth.
Surprisingly, We hardly saw anyone on the trails out there late June so it was really nice.
We have been going to mammoth for many years but only a couple times with the jeep so I have a lot of exploring yet to do. Your best bet is to stop by the visiter center when you get into town and pick up the latest four wheeling/ offroad maps. You should ask for the 2 free maps because neither one show all the trails, but between the 2 you can get around pretty well and avoid getting lost.
My last 2cents are to make sure to hit mammoth brewery and pick up some epic IPA's!! We also used to pick up a growler of root beer for the kids when they were younger. Shoot me a pm and I can shoot you over a pic of the maps I've got. I can't get my wol app to sinc with my iPad for pics.