Updated: 3:37 p.m.

House Democrats are refining their strategy for moving several languishing pieces of their jobs agenda through the chamber before the fast-approaching July Fourth recess.

Faced with continued inaction in the Senate on a broader package of middle-class tax extensions and jobless benefits, House leaders have added to Tuesday’s schedule stand-alone bills restoring unemployment insurance benefits through November and extending a first-time homebuyer tax credit.

“That does not mean we want to leave everything else on the table,” Majority Leader Steny Hoyer (D-Md.) insisted Tuesday. Hoyer noted that the House already has passed legislation extending unemployment benefits that expired June 2 but said the Senate has failed to act.

The House last week sent to President Barack Obama for his signature another smaller piece of the tax extenders legislation, which would restore cuts in Medicare payment rates to physicians.

Speaking to reporters at his weekly pen-and-pad session, Hoyer also said the House was on track to consider an emergency war supplemental “in the next couple of days, certainly by Friday.”

Given growing public concern about the administration’s Afghanistan policy and pressure from liberals who oppose the war, Hoyer said House leaders likely would hold votes on different aspects of the supplemental package. He urged Republicans to support at least the war funding portion of the measure. Democrats also are looking to tack on some domestic spending items, most notably money to stave off teacher layoffs.

Congressional Progressive Caucus Co-chairman Raúl Grijalva (D-Ariz.) said liberals had received assurances late last week from leadership that the war money would be pulled out for a separate vote.

House Democrats have started whipping Members on two separate questions: whether they would support money for the war and whether they would support the other domestic items, a Democratic leadership aide confirmed. The aide said the idea would be to structure the rule so that it would allow for two separate votes on the different aspects of the measure, which would remain in one bill.

Before leaving town, House leaders could try to spin off additional pieces of the stalled extenders package. Hoyer said it was “very disappointing” that the Senate had been unable to act on the broader bill, in part because it also includes money for summer jobs. He did not indicate any plans were in the works to break out that funding for a separate vote this week but said such a move was “possible.”

Whether that could get through the Senate is “unknown,” he said.