While I did not find the word outstaff in any dictionaries, a web search does turn up several companies named Outstaff or OutStaff, perhaps created as a portmanteau of outsource and staff.
In the U.S., the practice of long-term but indirect employment through a third party would be called contracting or hiring a contract worker. Even though direct employees may also have employment contracts, only third party employees are called contractors (contractor has many other meanings, however, so context is important); the contract is between the hiring company and the staffing company, contracting firm or employment agency which provides the workers.
A contractor may be distinguished from a temp, a temporary worker who might be brought in on a short-term basis for low-skill duties (e.g. basic clerical or reception work) while a regular worker is out, or who might be hired for a seasonal period. Contractors may also be distinguished from consultants, who are specialists temporarily hired for their specific knowledge in an area, as opposed to someone filling in for what would be an ordinary internal hire. The lines between a temp, a contractor, and a consultant can become quite blurry, depending on the specific employment situation, and relate as much to social status as to employment status.
When a job formerly handled by internal staff (whether employees or contractors) is assigned to an outside firm, the process is known as outsourcing or contracting out, and may be handled by an outsourcer or outsourcing company.