I'm noticing two things but I think they're in conspiracy.
I ride a Trek 4300, about 4 months old.
A month or two back I raised the post about 4-5mm (I could judge the distance because the rear reflector holder was fitted at the base) just to see how it fitted - I'd fitted at the shop and felt fine. I felt like experimenting (). It felt fine higher, so I left it...but about a week later I noticed the seat post was resting against the reflector holder again. I raised it 4-5mm again, and tightened the quick-release on the saddle a bit - it wasn't loose before, but it was well firm to close after the adjustment.

Around the same time as this, I'd noticed that sometimes when climbing hills, the bike developed an odd clicking noise somewhere underneath me. Could never figure out where exactly because it needs to be pedaled under strain to get to it. The sound is like a clicky pen, the kind that have the clicker on top - if you push it in so the nib is out, and then shake the pen you get a plasticky clicking noise, and that's what I'm hearing on the bike.

For the last few weeks I'd left it alone, and heard no noise. This week I raised it again, and the noise is back.

It would seem the noise is related to the post not staying up, but I don't understand how.

My question is - what can I do to make the post stay where I want it? Should I tighten the quick release more?

Post may be too small for the bike and doesn't stay put. Look at the slot at the top back of the seat-tube where the clamp-squeezes. Is it parallel? Or is the top of the slot and the clamp-ears squeezed completely together? Take a photo and post here if you can.

Maybe your seat post clamp is fine but your reflector is loose and falling down? That might explain the clicking noise as well, reflector bouncing on the frame...

I had a specialized mtb that the seat collar could never hold the stock seat post tight enough that it did not slip. I swapped out the seat post for a thomson and the seatpost collar for a salsa and it never slipped again.

You could try moving the reflector down to use it as a stop and see if that noise goes away. (if you already hadn't)

Okay, sorry about the delay, went for another ride today to make some checks after re-raising the post 4-5mm.
As soon as I heard the noise start up (on a hill, as always), I stopped and took off the reflector completely. Rode on, noise still there. The only other thing I could notice was that it's not matched by pedal stroke or the rear wheel (happens in random places of both rotations). The noise definitely goes away as I reduce effort in the pedal stroke. wtf

I think that was a good suggestion to check the roundness, DannoXYZ, although as far as I can tell it looks okay. Pic (I presume that's what you were talking about?) attached - reflector not.