Spa Panel with 50a breaker
Your #6 wire is good to 60A.  (Actually it's good to 55A, but they don't make 55A breakers, so you get to round up to the next available size). 

As such, there is nothing wrong with fitting a 60A supply breaker inside your main panel.  

The ruling question is whether your spa subpanel's internal buses are able to handle 60A.  Did they put a 50A breaker there to protect a spa they never bought? Or is it there to protect the subpanel itself?  Model numbers would help us research this.  

Separately, if it is not attached to the house, it will need a main disconnect switch or shutoff.  Usually the cheapest way to get a shutoff is to use a sub-panel with a main breaker (just using the breaker as a switch, not a breaker). 



So what's up with people telling you that you need to bump to #4 copper? (which you would never do by the way, you'd use #2 aluminum if you needed that, which you don't). They are probably up-in-arms about that because of the long distance, and voltage drop which results.  

However, voltage drop is the victim of an "old wives' tale", cheerfully amplified by the sales departments at wire manufacturers.  The wrong info is to calculate voltage drop on breaker trip current, and that 3% is condemning.  Wrong and wrong.  

Actually, calculate voltage drop on what the load actually is (if you are at breaker trip current, you have bigger problems).  And 3% is a rosy ideal; depending on application, 6%, 8% (Code starts objecting at 8%) or much more are perfectly acceptable.  And this is one of those applications where you have a lot of slack.  

So you don't even need to think about voltage drop for this spa below 250 feet of run.  (That's not universal. On some circuits it's a concern in as little as 100'). Even above 250', I'm only saying find a piece of paper to run some numbers.