What you're talking about is NOT a safe (or prudent or legal) thing to do. If you have no room in your box for an extra double pole breaker, there are basically two options:
1. Remove FOUR of the 15 amp standard service breakers, (sitting side by side) and install TWO DOUBLE 15 amp standard service breakers in their place. This will give you two breakers with TWO separate circuits on each. (They'll be the same size as the present ones, only each breaker will have two small switches on them instead of one). Something similar to this:
Dual Circuit Breaker
This way, you'll free up two side by side slots for a large main breaker running to the spa.
2. Call an electrician and get an estimate for a break out for a sub-panel.
I would never attempt to bypass the normal legal and practical methods offered by standard breaker options that will work properly in a panel system. Going directly to your mains (in the main box) is not only illegal, it is dangerous, particularly in the long term.
Hope this helps. I've used option number 1 quite a few times and it works well. Just be sure you ascertain the correct brand and type of circuit breaker that your panel requires prior to making the trip off to the supply store -- (take one with you!).
Lastly, I'm trying to figure out what spa requires a sixty amp circuit? If it really does draw sixty amps, then you're pushing it a bit with six gauge wire, especially of this length. I would consider having a pro do it the right way if this is the case.
Thanks.