Yes, you'll need to support the joist above. Construct yourself a temporary stud partition that, from top to bottom, is about 1/4" taller than the distance from the bottom of your joists to the concrete floor. Add cross-bracing on both sides to keep the wall square. With assistance, stand the temporary wall up against the joists a couple of feet back from the doors and hammer into a vertical position. What you're doing is slightly lifting the load of the joists above off of the header. Don't lift more than the 1/4" or you may damage existing construction and connections above. Once the load is off, you can remove the header and replace it. When the temporary partition is removed (the reverse procedure of installing it) the load and existing construction will settle back into the same position it was before. Not raising the load, even just a little, makes it quite difficult to get it back to where it was in the beginning.
Just a suggestion....if you can obtain a couple of pole jacks, you can do the same thing, but without having to build the temporary partition. You'll just need a temporary header between the jack heads and the joists above.