I'm going up to 265/70-17 Monday and buying the same size tire as a spare too.
I just looked and it's going to be tight.
There's a leaf bracket or whatever that is about an inch from the 265/65-17 spare.
A 70 series tire is 1.05" bigger diameter.
I asked this question on Tacomaworld and someone replied...
I have a 285/70/17 spare under mine...fits great. It's a little tight and it doesn't go all the way up, but its tucked enough that it doesn't stick out much. Also, there is a DIY to cut out two tabs to make more room. I didn't do that, but what I have works