When you are viewing your set, make sure that you have the set in Custom sort order since other sorting methods of the set cannot be rearranged. You should then be able to tap on the Edit button to enter editing mode and use the reordering handles to change the order of the set.
You can also utilize drag and drop to rearrange songs as well as organize your library. To do this, tap and hold on the song you would like to move until it "lifts" off the list. You can then move that song around while also adding additional songs to the drag session. Finally, drop the songs into their new position in the list.