Monopsony models generally imply they should, and that is part of the argument why minimum wage hikes might be good for workers, wages, and yes even employment. But the data don’t seem to support the claim of more search behavior:
Labor market search-and-matching models posit supply-side responses to minimum wage increases that may lead to improved matches and lessen or even reverse negative employment effects. Yet there is no empirical evidence on this crucial assumption. Using event study analysis of recent minimum wage increases, we find that increases to minimum wage do not increase the likelihood of searching, but do lead to large yet very transitory spikes in search effort by individuals already looking for work. The results are not driven by changes in the composition of searchers.
That is from a new NBER Working Paper by Camilla Adams, Jonathan Meer, and Carly Will Sloan.