Here in Ireland, my local posties are brilliant. We don't have a letterbox, just a cat flap but they're happy to post the mail through it. (We plan to have a proper postbox built into a new wall this summer).
When we first moved over here, we were renting in a village a few miles down the road. We then bought our current house and had an overlap of a couple of weeks in our rented place before we moved in to the new place (just before Christmas) a few years ago. We were pottering about the new house having some work done when the new lady postie introduced herself. My DH asked her about the forms for organising the mail re-direction and she said 'sure, you don't need to bother with all that, I know you live here now'. She knew the postie who covered the round in the other village and he gave her any mail that was addressed to us. Even 2 years later if anything is still addressed to the old place, it gets delivered to our new place, no bother.
Also, when we were in the rented house, a parcel was delivered but we were out and so the postie delivered it to the landlord's place (around the corner). We saw the postie a couple of days later and he asked if we'd had the parcel yet. No, we weren't aware of it. So, a little while later, he knocks at the door with the parcel. Turns out the landlord was out but the window to his conservatory was open and the parcel was on the windowsill, so the postie had reached in to retrieve it...!