Can someone answer a congressional procedural question for me.

If the house approves a bill and sends it to the senate but the senate does not bring it up for a vote and then there is an election, can the new senate bring the bill up for a vote and as long as they vote for it as is, can it then be sent to the president for signature?

What I am trying to determine is, if the house sends over a number of bills this session and the republicans gain the house but dems get a bigger majority in the senate and eliminate the filibuster; can they then pass the bill to go to Biden for signature?