I am trying to round over the edge of a 4ft long 1.5" x 3/8" strip of
hardwood using a 3/8" round-over bit.

Since the wood is so thin (3/8"), I am using my router table for
otherwise the bearing would extend below the wood (and in fact the
roundover diamter is the same as the thickness of the wood).

The problem is that I am having trouble doing so without causing jumps
& gouges in the edge.

The problem is particularly bad near the tail end where I get a rather big
jump when the strip leaves the infeed guide. I believe the problem is
due to the fact that when the entire edge is rounded over, only the
very bottom edge of the strip is touching the guide which makes
alignment very difficult.

So, I am looking for a reliable way of rounding over the entire edge
of a thin strip of wood as above.

Are there any tricks? (such as maybe temporarily attaching it to
another piece of wood)