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Do you mean your pecs, or the upper half of your pecs? Not that it matters, I think. Supposedly anything pushing down and out (like dips) mainly activates the lower part of the chest, anything up and out (like inclines) hits the upper, but I've always been a bit sceptical whether the difference is going to be very noticeable for most people. I think it's mainly genetic - depends where your body naturally has the tissue.
Personally the move towards lifting such heavy weights that I fail in 2-3 reps seems to be having useful effects. Bits of me are wobbling again for the right reasons when I walk down stairs. The added advantage is that there is nothing like the same pain involved. 3 chins with 88lb hanging off doesn't really hurt, 17 ordinary ones burns and seems to cause a lot more damage that takes longer to heal.
Reference the push ups in the handstand position - horribly difficult and something that a lot of people (me included) could never do safely. I think if you are vertical it is going to be shoulder and tricep rather than chest. If I had a smooth bit of MDF I would try putting that at the bottom of the stairs and try pushing myself up it. I've a feeling that would be about right for upper chest.
Should have been clearer about the handtand press. You shouldnt try to be vertical, but at an angle maybe 30 degrees from vertical. That way it focuses on chest and pecs. Isnt that difficult and not dangerous. Give it a try.