Yes, you have a full size spare.
And to go offroad alone isn't the brightest thing a person could do.
Water, food, blankets, flares, survival radio, shovel, matches, tools and I carry a gun.
Keeping a charged cell phone is OK, but most times, you're out of range.
Tell folks where you are going, giving good details, and letting them know when to expect you back. Call your contact person when you get back safely. Let them know that if you don't call them, to wait 24 hours, then call the BLM or the police to let them know what's going on, and where.
Plan to be wherever you get stuck (if you can't fix it on your own) for at least a minimum of 3 days! Possibly even a week.
Here in the mountains, we could roll over a cliff, and not be spotted by the casual observer for a long, long time. It's something those who do venture out on their own know about, and actually plan for. If you don't, you're in trouble.
Try not to go off on your own if at all possible. Sometimes, cabin fever sets in, I know. But the best thing is to be with someone. Hey, you never know, you might have to eat them!