There will be another bill but it most likely won't have another direct check to taxpayers. The focal point will be an agreement to provide direct aid to state and local governments and whatever the Republicans leverage that for. Current potential items on the list include a payroll tax cut and COVID-19 liability immunity for employers.
The House bill is just the Democrats' staking their ground for the upcoming negotiations, and they threw everything in. If Republican side has to burn chips to remove silly things like the Trump signature thing, they will have less chips to use on things that actually matter to the Democrats.