Look at the unit, removed from the guitar. There is a "lever spring" that holds the male plug in place. This can often be the cause of what you speak. Determine if you can (gently) bend it into place so that it makes VERY firm contact with the plug. While it is outside the body - take a plug and insert it into the plug body (female) and look very closely to see that it contacts well and is firm. IF it wiggles, especially from side to side (the spring unit) the plug unit will need to be replaced. If it's firm and your slight bending has made it so that contact is tight and wiggle free; you should have good results.
IF you still have trouble and it's a firm contact, experiment with an entirely different MALE unit (a different lead completely) - it may be that the male has some issues. If not then replace the unit and be done with it as it might be something like wear on the retention spring that has caused some roughness or wear, etc. Better to buy the part via a electronics shop then the "Fender" Custom whatever as utility is more important than any other thing and it's not something anyone will notice.
