Under contract on Triplex. One unit vacant. Can I try to fill?

[Josh Michel]

I'm recently under contract on my first investment property, a Triplex, one unit of which is vacant.  I'd like to try to fill the unit for as close to the closing date as possible, which is about 55 days out.

Is it OK to shop the unit (with the current owner's consent) with the available date as a few days after close?  I realize I cannot show the unit (unless I get permission to from the seller of course) until then.  Assuming I cannot show it, is it wise to try to fill it?  At least get the ball rolling?  What pitfalls will I hit if I do shop it?  Does a rental get "stale" if it sits on the market too long because I cannot fill it until after close?

What would the BP forums recommend?