WASHINGTON—Senate lawmakers turned their full attention to the Trump administration’s proposal for $1 trillion in spending to combat the coronavirus pandemic, including aid for airlines and direct payments to American households, after passing a paid-leave bill that President Trump signed into law late Wednesday.

The urgent work in Congress came as the coronavirus toll crossed a grim milestone, with more than 200,000 confirmed cases world-wide and growing job losses. Two lawmakers, Reps. Mario Diaz-Balart (R., Fla.) and Ben...