Yes, a more thorough cleaning includes a can of Deox-it and a small brush. You can get small brushes for cleaning between your teeth. They will work very well on the small tube socket.
Be certain the socket for the 6080 is snug and give it a squirt of the Deox-it then scrub all around with the same dental brush.
The tube pins can be brushed with 4/0 steel wool. Be certain nothing stays on the pins. The Deox-it in the socket will help keep the contact between the pins and socket good.
But it sounds, to me, like it is tube related.
Startup hum beyond 15 seconds is not normal. But if it is less, just don't listen at startup.