Can I use #8 wire in a 15 or 20A circuit?
You're always allowed to use bigger wire

The circuit ampacity defines the minimum wire size needed.  If you have larger wire on hand, go ahead and use it.   

For instance I often run single appliance circuits that demand a 15A breaker. However, I stock 10 colors of #12. I don't stock #14 at all because it's a useless and redundant wire size for me, I'd rarely use it. So the circuit gets run in #12 and breakered 15A.  It's a 15A circuit because it's breakered 15A. 

Good to be self-aware of the impulse to always match wire size exactly to the breaker.  That is a false impulse.  There are actually several reasons to consider an upsize besides the one I use: distance, conduit crowding, and yeah, use of existing wire. 

The splice must be accessible

One of the rules of splicing in conduit is the splice must remain accessible.  So there must remain a hatch cover which can be removed to get to that splice.  Even if there was not a splice, if it is a junction box or other pulling access, it also must remain accessible. 

If you wanted to bury the old junction site, you would need to reconfigure the conduit so it was straight thru, without any stops, to the garage.  Then I would see if I could shift the #8 wire so instead of going from spa to panel, it goes from the garage to a point still inside the house where you could fit a junction box.  Then extend the #8 from there to the service panel so the whole run is #8.  

Keep it metal

If the old conduit was metal, extend in metal of the same type.  IMC and Rigid conduit can be buried at 6" cover, which is pretty handy.  Otherwise you will be at 18". Those metal conduits are able to serve as the ground wire.  You need separate neutral and grounds, no bootlegging.