Not a bad plan. Good first design.
C1 is probably not needed.
Rule of thumb: cathode resistor is typically MUCH smaller than plate resistor. For the usual tubes, for good gain, usually 20 to 100 times smaller.
12AX7 is often run with 100K in plate, 1.5K in cathode. 67:1 ratio. 12AU7 favors more like 100K and 5K, 20:1. 6AU6 as a straight audio amp might be run with 50K plate and 1K in cathode.
That's for good gain. Here you want some dial-back. Increasing cathode resistor will do that. It also makes the tube more linear (less "tone"). And I wonder if you ever have need of the full 100K?? I think you want values below 0.5K to maybe 5K-10K. You can't conveniently set these teeny values on a 100K pot.
I'd get a bin of brown-stripe and red-stripe resistors. Try them all: what it sounds like with 100 ohms and 3.3K ohms in cathode. Find a pot which just about covers the high resistance case yet can be dialed-down to the useful (and more loud/toneful) lower resistances.
Conversely: raise the plate resistor. Too high is not great because R4 R5 and amp load it. However you might try 330K, maybe with R4 snipped for test.
Triode wiring is worth a try. Different. Perhaps not better, but who knows?
What is your plate voltage? Both extremes: R2 at 100K and at zero ohms? Plate voltages "should" be around "half" of B+. On 300V supply, 100V to 200V, but much lower here. True, there are reasons to do otherwise. But if plate is slammed to B+ or down near the cathode voltage, output is very small and dirty. Which is kinda what you want, but you want a zone that is not weak yet not too clean. Plate voltage is a clue.
Low-low-low-voltage operation is tricky. You have Screen grid G2 going to B+. You never do this at "real" B+ voltages in voltage amps (power amps, yes). You get more gain at lower G2 voltages. But 9V is already very low, so.... Still, you might try feeding G2 from your 6V tap, or even do the conventional thing with 220K B+ to G2 0.1uFd (or 5uFd 10V electro) from G2 to ground.
At very low voltages, sneak current in G1 is not negligible. Some experimenters have used much lower values for your R1. However this reduces input impedance which loads your guitar. Others have leaked some positive voltage at G1, perhaps 10Meg from B+ to G1 (now you better have C1!).
> There's some hum (not terrible)
Is it in a box? Tube-works out in the open DO catch ALL the hum in the room, annoying on low-level signal. Everything in the audio path should be in closed metal box. Tubes usually get shield-cans, though you can wrap with foil touching chassis for a test. If you do like Triode, there a trick: use G2 as your signal plate. Ground the designated plate. The plate is the outermost element. If grounded it shields all the other internal bits. G2 works fine as "a plate", up to some small amount of power. At 9V you can't possibly pass enough power in G2 to hurt it.
What do you have for B+ filter?
Are you SURE your LM317 is _clean_? Rough ripply 9V into a 6V regulator can give you ripply 6V "DC" which is a worse buzz than pure AC would be. 10 ohms to 2,200uFd would drop 9V down to 6AU6 heater level and be more-sure to be real clean.
When tubes are worked at VERY low B+ voltage, sometimes it helps to bend the specs on the heater. 7V or 8V in the "6.3V" heater will give some more plate current, and there's still many hundreds of hours of heater-life. However high heater voltage may excite more G1 leakage and force current down, you could try 5V heater.