Go message oriented.
If you choose messaging as the strategic approach and refactor (if necessary) business logic around JMS, then other options remain viable and locally applicable.
With this approach, you pay a specific fixed cost (refactor) in your existing (single tenant) system. You then can apply approaches of various degrees of complexity, ranging from simple sharding (@Geziefer's id based association) to a full blown shared-core-schema + extended-tenant-specific-schemas approach, without impacting system architecture and additional refactoring.
You will further have orthogonal control over your system data flows via the messaging layer (applying routers, filters, special processing paths, etc.)
[edit per request]
There is nothing per se in M.T. that explicitly suggests message orientation. But as a general problem, we are looking at widening interfaces, and enriched data flows. Per an API based approach, you would need to carefully inject the appropriate the tenant discriminant in all required interfaces (e.g. methods). A message based (or alternatively a context based API approach) allows for a normative (stable) interface (e.g. message.send()) and at the same allows for explicit specialized data flows. If switching to a message based backbone is not on the table, you are strongly suggested to consider injecting a uniform context (e.g. "RequestContext") param in your APIs. This single extension should cover all your future specialization needs.