Should I sign with one recruitment agent or several?

If you're working as a locum, you are not obliged to sign with just one company.

In fact, it generally makes sense to sign with more than one initially, because no single agency normally has access to all the jobs that may be available.

By working with a range of agencies, you'll often find one or two that suit your needs better. Many doctors who locum for a while eventually settle on one particular agency as their preferred agent.