Generalized Pustular Psoriasis (GPP): This is the most likely diagnosis given the presentation of a widespread pustular rash, fever, and a history of several weeks of illness. GPP is a rare but serious form of psoriasis that can present at any age, including in older adults, and is characterized by the sudden appearance of tender, sterile pustules on a background of erythematous skin.

Acute Generalized Exanthematous Pustulosis (AGEP): This condition is characterized by the rapid onset of numerous small, sterile pustules on a background of erythematous skin, often accompanied by fever and systemic symptoms. It is usually triggered by medications or infections.
Pustular Drug Eruption: Similar to AGEP, this condition involves the development of pustules due to a drug reaction, which could explain the feverish state and improving rash if the offending drug has been discontinued or its effects are waning.
Bacterial or Fungal Infections: Certain infections, particularly those caused by Staphylococcus aureus or Candida, can present with pustular lesions and systemic symptoms like fever.

Septicemia or Bacteremia: Although less likely given the improving nature of the rash, it's crucial not to miss a systemic infection that could present with fever and a rash. Conditions like meningococcemia or streptococcal sepsis can be rapidly fatal if not treated promptly.
Toxic Epidermal Necrolysis (TEN) or Stevens-Johnson Syndrome (SJS): These are severe skin conditions usually triggered by medications, characterized by widespread skin necrosis and detachment, and can present with fever and systemic symptoms. Early recognition is critical due to their high mortality rates.

Subcorneal Pustular Dermatosis (SPD): A rare skin condition characterized by the formation of sterile pustules just beneath the stratum corneum, often accompanied by systemic symptoms like fever. It can be associated with various underlying conditions, including inflammatory bowel disease and hematological malignancies.
IgA Pemphigus: A rare autoimmune blistering disease that can present with pustules and systemic symptoms, although it is more commonly associated with blister formation.
Neutrophilic Dermatoses: Conditions like Sweet syndrome (acute febrile neutrophilic dermatosis) can present with fever, systemic symptoms, and skin lesions that may resemble pustules, although they are typically more papular or nodular.