They switched from 245/75/16 @ some point to 255/70/16. it's basically the same size tire, or they are considered as "plus zero" size tires in comparison. The side wall height is a percentage aspect of the tread width (sidewall height is 60% of the tread width in the latter example), so 65% of 245mm is similar to 60% of 255mm.
And yes, I also have a full size (245/75/16) spare under my truck. It will fit.
Also, might be a good time to mention for people to check the air in their spares. I never had mine down (thank you BF Goodrich A/T's) and the spare had like 6 psi. Wouldn't have done a whole hell of a lot of good.