The hot tub is wired to a sub-panel, but there is a breaker in your main panel where the sub-panel connects. Is that a 60 amp breaker, or is it larger? Usually you want a breaker rated for full load plus about 10%. If its a 60 amp breaker, then at 54 amps you are probably running it at the max planning load.

Bruce

I disagree with the calculations. You do not just size up feeders by 10% fo the load. You use 125% of the continuous load and 100% of the noncontinuous load. The size of the breakers installed in the subpanel is spurious.

The spa is not a continuous load and given the wording in the original post, I don't think the 15A circuit being added is either. You can go up to the ampacity of the feeder and its breaker. No need for fudge factors as far as the code is concerned.
You will need to identify the breaker protecting the feeder in the panel (and make sure that the conductors between the two panels) are sufficiently sized, but I can almost bet it is also 60A for both.

Thanks to those that have replied.

The breaker at the main panel is a 60 amp breaker as well. From the main panel to the hot tub panel the feed is #6 wire and the distance from one to the other is only 6' total wire length. The additional circuit will not have a continuous load either, it will be used for an out building with shop light.