SOLUTION:
1) Double check size of cap and artwork and adjust appropriately prior to digitizing.  Different sizes of caps are available, and knowing the style prior to the digitizing is important.  If knowing is impossible, it's best to stay at a size appropriate for a low-profile style, approximately 2" high.  Average cap height is 2 1/4" high.

2) Use as little underlay as possible, but USE IT!
One line of running stitch will usually suffice below columns unless it's for a fabric like wool.  For a 6-panal cap or other that contains a front seam, begin the design by filling in the "ditch" with one or two lines of running stitch, followed by a light density zigzag stitch.  This should help keep top stitching from falling into the seam.

3) When digitizing for fills, avoid a horizontal angle.  This causes stitching to pull inward on the sections sides and the stitching is already fighting against the curved shape of the cap.

4) Create the design so that it sews from the bottom up and center out.  This isn't always possible, but staying close to the rule will help retain registration.