Besides all the great gear you always carry, do your Element a favor and perform maintenance before it's due. My "more than 8 year old" Ford started overheating and when checked out the radiator was 90% blocked -- the cost was a new radiator core.

Most all of the things that can go wrong have been replaced with new from the rear axle to the A/C. The serial number is old but the body is solid and everything that makes it move has been replaced. I'm so far ahead on $$ with this truck I try to replace stuff before it breaks. In most cases the cost of a tow or down-time car rental would exceed the cost of early replacement.

To answer your question though, yes, when I leave SOCAL or go anywhere of increased elevation like the mountain passes in southern Oregon (way outside SOCAL), I have a winter kit duffel bag that goes along for the ride which contains:
Winter parka
Smartwool long u/w
Wool & Fleece sweaters
Nylon outer pants
Couple pair of Smartwool socks and sock liners
Boots.
A couple different beanies, wool & fleece
Wool & fleece blankets
MSR MicroRocket
Canned food & water are carried separately.

That's a partial list and I usually dress for the climate I'm in so the list of what's in the duffel closely matches what I'll be wearing as I drive.

I'll be fine, mostly because I'm fairly sure the truck will not break down. There's always the chance of a random accident though and the saying S**t happens still applies.