The Bhoys were eliminated after losing 6-1 on aggregate to Legia Warsaw, before being reinstated after the Polish club were disqualified for fielding an ineligible player.
Ronny Deila's side now face Maribor knowing that victory in the two-legged tie will see them qualify for the group stages, and Mulgrew says that a sense of embarrassment will not hinder the Scots.
"We are grateful to get another chance," he told reporters. "It is not the way we wanted to get through but we have to get on with it.
"It was disappointing how the two games went but rules are rules and we are happy to be back involved.
"Maybe there is a wee bit [of embarrassment] but what can you do? You have to quickly get over that and get on with it. You can't feel sorry for yourselves and be embarrassed at getting through."
Wednesday's first leg in Slovenia is scheduled to kick off at 7.45pm.