If I use the Montgomery GI Bill® first, then I can also use the Post 9/11 GI Bill®, right?

It is possible. If you’re authorized to use more than one GI Bill® benefit you’re allowed to use up to 48 months of benefits, but one at a time. If you have used up your entire chapter 30 and have never used a different chapter (31, 32, 34, 35, 1606, or 1607) you could be eligible for an additional 12 months if you have not reached your delimiting date.