W/O knowing how old the 6' tree is (or what kind it is), I recommend against moving it until the Fall. This way, by Fall the tree goes dormant and won't be stressed too much from the move, and will wake up in a new environment ready and raring to go next Spring. If you move it now, you may kill it. Nothing is for certain but you MAY kill it if you move it now. (You can find out from a nursery. Bring a leaf w/you so they can properly identify the tree too.)
One possibility is to, temporarily, work around the tree. Pretend it's not there and simply don't install the section of fence where the tree is. Mark a fence post location on either side of the tree (as you plan in your layout) but wait for now to dig. Dig and install all the other fence posts now and put them in. When the Fall comes, move the tree and dig and install the 2 remaining fence posts, and hang the 3 remaining sections of fence. (Draw your posts and tree on paper if you can't envision it.) So, if your fence posts need to be 10' apart (on center), then the 2 fence posts CLOSEST to the tree should be 30' apart (on center). After the tree is moved, the 2 remaining posts and fence can be installed.