That is also what I do, usually.
My method for getting the top off involves setting the TS blade height to just slightly below the side thickness. Use the box next to the blade to get the height about 1/32" below thickness.
I then run all four sides.
Next I'll bump up the blade for a through cut and run the long sides through. This leaves the lid still firmly attached on the ends so no surprises while cutting.
For the lid removal, I then use a box cutter. The blade is inserted into the kerf and worked to sever the last little thinness of wood. Keep the blade angled so each cut is pulling from the inside of the box out. This will prevent chipping out on the box inside.
Once the lid is free, I'll use a scraper at the same angle to remove the little lip (work from the inside out). Then I'll go to a sheet of sand paper on a flat plate and go at it.
This gets me a perfect flush fit along the perimeter.
There are a lot of other techniques for cutting, one being to placing shims into the saw kerfs as you cut the last two sides. This keeps the lid/box in position and if done right, will avoid any saw blade "kisses" on the edges.