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Mine is not a CC, but it has 6 valves. Unfortunately you didn´t mention the length of the 5th.
So my horn has a halfish and a fullish step 5th and 6th valve.
Each horn is different, so it´s up to quite some work with the tuner to figure out fingerings.
I didn´t own a tuner when I got started on that horn, so my approach was this: will I need little, medium, or flat out BIG compensation for a given note when played with 1 through 4 ?

Consult a tuner and find out what works best on YOUR horn.
Examples:
I figured 2-4 would be close enough, and 1-4 would be the first to need compensation. 4-5 was still too sharp to be called a BBb on an F tuba. So I settled on 1-4-5, lipped up a little.
From then on I just used my right hand as if I had a 4-valved tuba and thought: "4-2: no comp. 4-1: little comp (5); 4-3: little comp (5) 4-2-3: medium comp (6); 4-1-3: medium comp (6) 4-1-2-3: maximum comp (5,6).

This probably is NOT the mathematically correct way, but it was logical. Quirks could be lipped easily, and this pattern faciliated fast passages, since the right hand gets to do the same throughout the range.

The 5th valve on my Yamaha Eb is a 3/4 step valve. It comes in very handy in the low register. Since it is dependent (within the 4th valve loop), it's use is rather restricted. I use it for low Ab (2,4,5 equivalent to low F on a CC), low F (1,3,4,5 equivalent to low D on a CC) and low E (all) which is too sharp to be used. I believe that you'll find it useful where you may be able to use it as a register valve in conjunction with the 4th valve to play some low octave notes with their upper octave fingering adding 6 & 4 to the combination. You'll also find that using it will eliminate the need for some slide pulling that you'd have to do with a 4 or 5 valve horn. Sounds like it will be fun to figure out and use.

You are probably one of only a few people in the world to have a 6 valve CC. That said, there are a lot of 6 valve F tubas out there.
Every horn is different, so don't be afraid to try every possible combination especially in the low register to get a note that is in tune.
A few other places you may want to consider the 6th valve would be using 6-4 instead of 2-4 and perhaps 6-1-2 or even 6-3 if you had a 2-3 combination that was noticeably sharp. 6-2-3 could even substitute for 4.
On my F tuba, I play my low B (your low F#) 6-4 and all my C#'s (your G#) 6-1-2 instead 2-3.
There is a lot of good advice so far in this thread, but remember that every horn will have different tendencies, and certain mouthpieces can increase or decrease those tendencies.
Have fun experimenting; you may find you have a half dozen available fingerings for some notes!