An alternative using drivers and fixed offset of path.
Got into this answer, got sidetracked, got answered, got accepted will post anyhow
For walk cycles often like to use drivers and fixed offset of follow path. This will make switching paths simpler.
First recommend to Apply scale to your path
Ctrl A Scale This will make your scale unity and not warp the spide. Often strange behaviour filters from non uniform scale. If there is no reason not to have unit scale, make objects unit scale.
Set location of spider to (0, 0, 0), making the follow path constraint place it at start of path. Similarly to scale. If something is going to follow path using offset method, without translation from the path to have its origin to be at one end of path at 0, and other at 1.
Forward of the spider is -Y axis.
Driven the offset of the follow path constraint (note it is not made the parent in this case). Can add other curves and step to next path animating constraint influence
A simple driver to make the spider "walk" the whole path in 250 frames Note: Could keyframe instead
frame / 250
can type this directly into field with no existing driver using #frame / 250 The has tells blender to make expression a driver. Else simply edit directly in expression box.
Note: I'm using alpha version 2.83.8, and for some reason this only works if curve radius is checked in modifier. Didn't look too deeply into why. Believe in theory this should work without, may have something to do with the IK of the rig.
Very cool spider and rig btw.