Cellulitis: Given the patient's recent hospitalization and IV site at the location of pain, cellulitis is a likely diagnosis. The presence of warmth, redness, and inflammation on the dorsal hands supports this diagnosis. The use of antibiotics such as doxycycline and Augmentin also suggests that the healthcare team is considering an infectious etiology.

Phlebitis: Inflammation of the vein, potentially related to the IV site, could cause pain, warmth, and redness in the affected area.
Contact Dermatitis: An allergic reaction or irritation from the IV site dressing or other substances could lead to dermatitis, presenting with similar symptoms.
Superficial Thrombophlebitis: A blood clot in a superficial vein could cause pain, redness, and warmth, especially if related to the IV site.

Necrotizing Fasciitis: Although less common, this is a life-threatening condition that requires immediate attention. It presents with severe pain, swelling, and redness, often with a history of trauma or infection.
Sepsis: If the infection is spreading systemically, sepsis could be a concern, especially given the patient's recent hospitalization for aspiration pneumonia.
Osteomyelitis: Infection of the bone, potentially related to the IV site or a complication of the pneumonia, could cause pain and inflammation in the hand.

Erythema Multiforme: A skin condition characterized by target lesions, which could be triggered by medications like doxycycline or an infection.
Complex Regional Pain Syndrome (CRPS): A chronic pain condition that could be triggered by trauma or infection, presenting with severe pain, swelling, and changes in skin temperature and color.
Gout or Pseudogout: Although less common in the hands, these conditions could cause sudden onset of pain, redness, and swelling, especially if there is a history of similar episodes or kidney disease.