Go with the tubes first. Always keep a spare set, replace when you suspect a problem, if that fixes the problem throw the old tubes away and buy a new spare set.
What you are describing, I would bet, is the 6SN7 heaters are not powering up together (one tube is heating up faster than the other) and that imbalance creates the hum until both tubes reach operating temperature.
If new tubes cause the same humming problem the next thing to check is the tube sockets and the wiring connection to them.. Sometimes tube sockets lose their grip as tubes get pulled out and reinserted. That would be would be where I start when I open the hood. The next thing I check (on a cold amp) is the continuity of the RCA jack grounds with the power inlet earth ground.
If that does not fix it, then you start swinging for the capacitors.