As @Wiener_Demeanor said the multi meter is using zebra strips and yours is the more problematic pinned version. Have a look at the solder job on the pins but 99.8% of the time the issue is the pin to the glass connection. They actually lay down a conductive path of aluminum (vapour deposition) across the glass to the pin. You can see the tracks if you angle the light just right.
Typically the problem is the corner pins (more mechanical stress). I used to repair these years ago with a conductive epoxy (crazy expensive). Have a look around you might find something suitable.
Cheers
G