Will I kill my tree if I replant it about 6 inches deeper then the current depth?

It shouldn't under normal conditions. If there are still live stems above ground it would be pretty much like an airlayer. Figs are hard to kill if the soil is moist but not wet. You could likely bury everything 6 inches below ground and it would still push up a shoot.

Be careful of doing that. Sometimes roots that have encircled the root ball when the tree was in a container will continue to encircle the base of the tree trunk below the surface of the soil line. That is one of the reasons why it is recommended that when you plant a tree in the ground, that you plant the top of root ball even with the soil line or even several inches above the soil line so that the risk of girdling is avoided.

i plant my trees below the frost line, which is 12'' here. i have not lost a tree to winter since.
others, like pons,  have planted as much as 4' deep, slowly filling in the hole, with great success.
pons plants trees to last hundreds of years.

Thanks everybody for your advice and comments. From what I've read here, I'll go ahead and plant deaper then the current depth of the tree. In addition, I'll try to untangle any curled up roots.

I always plant as deeply as I can.  The only time that's a problem is if you hve a high water table and the roots rot.