The upgrade trick will improve the speed of your hut's being upgraded. Note that when doing this, the red bar (upgrading bar) will not move any faster, it will simply make braves not waste many seconds on finding and cutting the tree that will be needed for upgrade. Doing this can and will deny you many seconds or minutes of time wasting.

To upgrade a hut, send at least 3 wood piles to its door. When the upgrade bar reaches the top, the braves will have all the required wood there, and the hut will fastly upgrade. If there was no wood there, a brave would scout for 1 wood pile at time, meaning he would waste many seconds. Doing this to many huts will save you minutes of time.

It's usually good to upgrade with 3 braves. Select the braves, hold Ctl and click on a tree, and then on the hut. They will send wood piles from that tree to the hut. Now the hut is upgraded. Repeat this process as many times as you can.