  We define an invariant of contact 3-manifolds with convex boundary using
Kronheimer and Mrowka's sutured monopole Floer homology theory (SHM). Our
invariant can be viewed as a generalization of Kronheimer and Mrowka's contact
invariant for closed contact 3-manifolds and as the monopole Floer analogue of
Honda, Kazez, and Mati\'c's contact invariant in sutured Heegaard Floer
homology (SFH). In the process of defining our invariant, we construct maps on
SHM associated to contact handle attachments, analogous to those defined by
Honda, Kazez, and Mati\'c in SFH. We use these maps to establish a bypass exact
triangle in SHM analogous to Honda's in SFH. This paper also provides the
topological basis for the construction of similar gluing maps in sutured
instanton Floer homology, which are used in [1] to define a contact invariant
in the instanton Floer setting.
