If it is a dovetail joint, I am not sure what you would be gluing if you pushed the heal in place and glued. There would be a gap on the joints glue surfaces. Hide glue does not fill.
I have had that on a bolt on neck. Then I tighten the back bolt.

Terry if you are the original owner or you have a client who is the original owner please consider advising them, or yourself, to take it to an authorized Martin repair center. This is pretty easy stuff for the folks who do this every day AND short of using authorized folks for warranty covered repairs it could result in voiding the warranty for future, important work. Not that this is not important but if it's what I think it is it is not unusual and can be fairly easily fixed and in accordance with Martin's specified methods as well.

I checked out the pics and can't tell what model it is from the pics. I'm going to do something that I don't like to do and that is take a leap of faith here... that this is a 15 or 16 series with the single bolt, glued mortice and tenon joint. They do come loose from time to time and the proper way to repair these is very much not unlike a neck reset in that the neck may have to come off, the tenon glued in place again, the bolt tightened down, and the cover plate cleaned of original double stick, repopulated with new double stick and the cover plate replaced nice and straight and true. There is another technique where the neck does not have to be removed but then you don't have an opportunity to glue the tenon in place as the joint is engineered to have done.

It's pretty unlikely that a dovetail properly fitted... would come loose like this with a glue failure. After all a dovetail as eloquent and effective as they are is specifically engineered to have the shape of the joint and resulting mechanical attachment do the heavy lifting - not the glue.... The glue is there to 1) keep it all in place and 2) provide Luthiers with yet another way to lose our souls attempting to get the glue to soften up...