No one can say until it is opened up and assessed. Here is what the calendar mechanism looks like for a Felsa 690 series. Your problem may be a broken wire spring K' which pushes back on the pusher, or it could be spring G' which holds the month jumper in place. Or it could be just dust and dirt over the years that is restricting free movement of the pusher K. Or it might be something else, no one knows. These aren't that complicated but a repairer needs to know how to properly adjust them so they function correctly. You have to assume a watch this old will need a complete going over and adjustment.
Pic is from the book
Modern Calendar Watches by B. Humbert, in this case 'modern' means 1954. It's a very handy book I picked up years ago, it covers most of the calendar movements in use at that time and how to adjust them.
