Cellulitis: Given the patient's recent hospitalization and current antibiotic regimen (Augmentin and Roxy), cellulitis is a plausible diagnosis, especially if there was a previous IV site on the dorsum of the hand. The warm, red swelling is consistent with a bacterial infection of the skin and subcutaneous tissue.

Phlebitis: If an IV was previously inserted in the hand, phlebitis (inflammation of the vein) could be a cause of the swelling, especially if the patient has been immobile or has a history of venous disease.
Abscess: An abscess could form at the site of a previous IV insertion or as a complication of cellulitis, presenting as a localized, warm, and tender swelling.
Allergic Reaction: Although less likely given the antibiotic regimen, an allergic reaction to the medications (Augmentin or Roxy) could cause skin manifestations, including swelling, redness, and warmth.

Necrotizing Fasciitis: A severe and potentially life-threatening infection that requires immediate medical attention. Although rare, its presentation with severe pain, swelling, and skin necrosis makes it a critical diagnosis not to miss.
Compartment Syndrome: Increased pressure within the compartments of the hand could lead to severe pain, swelling, and potentially irreversible damage if not promptly addressed.
Deep Vein Thrombosis (DVT): Although less common in the hand, DVT could occur, especially in patients with risk factors such as recent immobilization or surgery.

Erythema Elevatum Diutinum: A rare skin condition characterized by red, swollen, and tender lesions, often associated with infections or autoimmune diseases.
Sarcoidosis: A systemic disease that can cause skin lesions, including swelling and redness, though it would be an unusual presentation for sarcoidosis to manifest solely as hand swelling.
Angioedema: A condition characterized by rapid swelling of the skin and mucous membranes, which could potentially involve the hand, though it's more commonly associated with facial or laryngeal involvement.