Staff surgeons could indeed be appointed to hospitals (regimental or otherwise) but according to Howard the role could be in the field:
“The designation ‘staff surgeon’ was applied to all surgeons who did not belong to a particular regiment. Thus it included surgeons employed on the staff of a general in the field, in a general hospital, or in a garrison. Staff surgeons were chosen predominantly for their surgical skills and were often promoted from hospital mates, apothecaries and regimental surgeons, although significant numbers came straight from civilian life.
During the Napoleonic period it was a war-time post only, and in peacetime staff surgeons were removed from the active list and went on half-pay. … the post could include administrative duties in the field, for instance acting as senior medical officer to a brigade or division. The hospital attachments were usually entirely surgical in nature. Staff surgeons were more often on foreign service than based at home. In 1807 of sixty staff surgeons all but seven were abroad.” n.17
Dr Martin Howard, Wellington's Doctors: The British Army Medical Services in the Naponeic Wars, (Spellmount Stalehurst, 2002) p. 10 citing
Drew, R. Commissioned officers in the medical services of the British Army, 1660-1960, Vol I (A Peterkin; W Johnston), London, 1968
Cantlie, N, A history of the army medical department, Edinburgh, 1974