If I earned a MOVSM at my unit, and then again while attached at BOLC, will I be eligible for the second award?

Yes ... theoretically. If you're attached for administrative reasons (sounds like you are going "TDY and return" to BOLC) then you can be given an award if you meet the eligibility criteria. As you will be attached to a different unit than the one that approved your current MOVSM, you would satisfy the "once per unit" stipulation.

However, I said 'theoretically' because your qualifying community service must occur while you are at BOLC and, unless you are already plugged into volunteer roles and/or the leadership just rubber stamps a recommendation (i.e., doesn't hold it to a standard of "exceptional, sustained volunteer service in the civilian community"), it is unlikely that you would be able to do enough to satisfy the subjective evaluation of leadership regarding your "overall level of volunteer participation" and the "impact of an individual’s community service".