We use the contact invariant defined in [2] to construct a new invariant of
Legendrian knots in Kronheimer and Mrowka's monopole knot homology theory
(KHM), following a prescription of Stipsicz and V\'ertesi. Our Legendrian
invariant improves upon an earlier Legendrian invariant in KHM defined by the
second author in several important respects. Most notably, ours is preserved by
negative stabilization. This fact enables us to define a transverse knot
invariant in KHM via Legendrian approximation. It also makes our invariant a
more likely candidate for the monopole Floer analogue of the "LOSS" invariant
in knot Floer homology. Like its predecessor, our Legendrian invariant behaves
functorially with respect to Lagrangian concordance. We show how this fact can
be used to compute our invariant in several examples. As a byproduct of our
investigations, we provide the first infinite family of nonreversible
Lagrangian concordances between prime knots.