two appliances on one 220 breaker?
First, a ground wire and neutral is mandatory for this setup. You cannot set up groundless, and you cannot "bootleg ground" off neutral as once was legal for dryers.  This is extra important because if a neutral wire has a problem, it will electrify the grounds - and that'll shock and drown people in the hot tub. 

If the hot tub already requires a 30A breaker, you should convert it to cord-and-plug connection.  Use NEMA 14-30 and extend the circuit to serve both 14-30 receptacles.  Why?  No receptacles are allowed if any hardwired load is more than 50% of circuit capacity (as the hot tub certainly will be). BUT multiple receptacles are allowed on a 30A circuit.   Of course, running both at once will trip the breaker, but you know that. 

If the hot tub uses a larger breaker, you must install a subpanel. However you will not need to upsize the wire run, unless you want to.  

If you go the subpanel route, don't be tempted to install one that's "just large enough". That's how your panel got full in the first place, which is why you have this problem now. The cost differential for a larger panel is tiny a couple of pizzas.  You often hear me say "I hope the guy who bought that panel really enjoyed that latté" because it's that cheap, so it's a waste not to do it. Install a large subpanel (in terms of number of spaces) - a 30-space is not excessive. Then as convenient, move some existing circuits to it to free up space in your main.    A house should have a number of spaces in the 40's, so if you have a 16, a 30; if you have a 24, another 24.