Currently there are options only for "component" and "version".

However, for a large team, it would be very useful to also have an "assignee board" (filtered by group so that the assignable list isn't too large), where the assignees are on the right. This would allow for team leads to see how many points each team member has assigned to them at a glance and also quickly reassign tasks by dragging to them.

This is the top requested feature for Jira as a whole by our team.