What’s the best way to adjust a leaning gate post

Last summer we rebuilt our fence using advice from The Fence Bible. The advice was great, and the directions clear. The author covered both concrete and compact dirt/rock footers for wooden fences. I'd highly recommend it to anyone building a fence for the first time.

Concrete does risk rotting the wood because of water build up, but that can be mitigated with proper drainage. Dirt footers can be problematic if you have very loose dirt in your yard. Dirt requires lots of work with something heavy to compact it; concrete requires mixing and handling concrete.

We set ours in dirt/gravel and realized it was exactly the right choice when our fence was hit by a car 4 months after we built it. The car shattered two 4x4 posts, a gate, and several rails, a 3rd posts was pulled from the ground, and every other post along the run tilted about 5 degrees including a 6x6 corner post; 10 posts had to be reset in all. Our neighbor (whose fence was also hit) had concrete footers, which were pulled from the ground, or shifted several inches but left in the ground. His footers had to be removed and replaced, and all new posts were required. We had to replace the damaged parts (2 posts, a gate and several rails), but we could reset the others. If we'd had concrete I suspect more would have broken, and we would have been unable easily reset any that had tilted (granted fewer might have tilted but that energy had to go somewhere, hence the assumption about more broken posts).

Don't attach the fence to the house. Aside from putting holes in your siding (not a good thing), your fence and your house will most likely move differentially. The attachment could result in a tear of the siding, which would be a bad thing.

Place the fence post close to the house, then run the fencing top and bottom supports up to within an inch or so of the siding (do leave a little gap, for that differential movement). Place the slats on the supports, allowing the slats to get just a bit closer to the siding.

You could even install a flexible membrane, similar to a product used to seal the bottom of a garage door at the floor, between the fencing and siding.