There are really only 2 scenarios:
1) You hire an agency and expect to work with them.
2) You hire an agent and expect to work with them.
If an agent leaves the agency, the only question that remains is whether the client(s) are under contract to keep working with that agent, or if they are under contract to keep working with the agency.
Most agencies, for various industries, are simply professional associations that bring nothing besides their name and their group of agents to the table. The agency, itself, has no contacts, and they really have no clout beyond the brand recognition their name engenders.
Here in Los Angeles, for example, a movie star might be under contract with Creative Artists Agency. Under the terms of that contract, they may end up being represented by an agent who arranges for work for the client. Of course, CAA can't arrange for work ... it's a corporate entity. The agent, herself, wouldn't even have any access to the client if they were not working for CAA, yet the agent probably maintains their own industry contacts which they mine to provide work opporunities for the CAA clients who are assigned to them.
When the CAA agent leaves the company for, say, William Morris Agency, the client's contract stays with CAA, and CAA then assigns another agent to the client. If the client doesn't like that arrangement, then they can (often) buy out their contract in order to sign a new one with William Morris, and then to continue to be represented by the agent who jumped ship.
If Hays' agreements with their clients are actually contracts between the clients and the agents, then Hays has not got much to complain about. They provide business services to their agents for a percentage of the contract's worth and the agent secure the contracts and do the work. The clients are not, in fact, Hays' clients.
However if the contracts are between Hays and the clients, then the agents can not simply "steal" the clients without the client first compensating Hays for the remainder of their contract. Any client who decides to continue being represented by an ex-agent would need to shell out some dough to buy out their Hays contract before they could sign a new one with the ex-agent, or the ex-agent's new company.
If Hays failed to protect themselves, then I have not got much sympathy for them.
It's not as simple as a bag checker at a local grocer opening up their own grocery nearby and siphoning off customers from their old employer, and Hays should be aware of the differences as they relate to their industry.