This is a completely normal situation to be in as a new joiner, especially in downtime periods.
As a new joiner getting a project is difficult as you haven't consistently proven yourself to the teams and staffing yet.
Additionally from a staffing or project management perspective getting an intern or an entry-level consultant would give approximately the same deliverable output but it would be much friendlier on the project budget (intern billing is free).
You did mention that you had a mixed performance on your first case but you should know that expectation for your first assignments is minimal and usually quite bad for most people. The good thing is that you are aware of this and you have discussed the feedback with your previous team.
What you should focus on is:
- Showing a good attitude, willingness to work, and that you are coachable
- Delivering on your assignments (whether it is a proposal, viewpoint, or project)
- Networking as much as possible within the firm (go to the office, mingle with people, and offer to support them if needed)
- Asking your first team and career manager for support in introducing you to other teams, and potentially referring you to new projects
Being unstaffed for a couple of days/weeks in between projects is normal for a new joiner. Just make sure you step up and deliver on the given assignments and keep track of your billability as you wouldn't want it to decrease too much.