There is a very small cut 1-2 millimeter or less long in the inner-tube, right next to where it's the bottom of the Schrader valve. The cut is only 1-2 millimeters far away from the valve's bottom.
The hole has a shape of a cut, parallel to the valve's edge, and it's very close, so it's very obvious likely that the valve was part of the reason (the other reason being the valve moved a lot probably because the inner-tube was not inflated.) Air escapes out of the hole so it needs to be repaired.
I don't know if I can put a normal patch, because if the patch's edges detach even only a millimeter from the inner-tube, the hole will no longer will be covered and air will again escape. Any idea? I give my 2 ideas:
Take a bike patch and cut a round hole in the middle, and then patch the inner tube with it, as if it were a ring, hugging the valve by all sides.
Get some super glue or cement that covers the hole somehow.