Last spring, when “Late Show” receptionist Nancy Agostini needed to find an apartment, her boss, David Letterman, lent a hand — on the air. Now, with winter just weeks away, Agostini needs a new coat. So Monday night, the talk show host, accompanied by nearly 30 staffers, took over the Saks Fifth Avenue flagship after hours to buy her one. After posing as a mannequin and burying the receptionist under a mountain of coats, Letterman bought her a $450 navy wool double-breasted number from Calvin Klein. The taped segment will air in a couple of weeks.