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Check everything:
Neck joint bolts
Bridge screws
Spring behind E-string saddle
Screw on back of tuner peg [already posted above]
Make sure nut is not shifting side to side
Check the peg slot for edge damage ... burring that
might be injuring the wrappings on the string.
Always put the right angle bend into the string [for
going down-the-peg-hole] BEFORE you trim off the
excess length.
Wind only about 2 turns of string onto the peg.
By odd chance, maybe you have proper neck relief
with no tension on the truss rod [due to summer
humidity, that could happen]. If so, take the slack
out of the truss rod [snug it up a bit].
FINALLY, quit checking the E-string tuning via the
tuning meter. Is it really going out of tune ? You
'quote' us your meter's opinion, but no mention of
checking relative tuning [string-to-string] by ear.
E-strings are a bit of a challenge for most meters.
Use the meter to tune the D-string and then tune
the remaining strings without the meter [by ear,
checking for 'beating' effect].
A bass is a simple machine. The user is a far more
complex machine. There's very little can go wrong
with the simple machine ...... and make sure your
"Bass and Guitar" tuner is set to "Bass". It will be
erratic reading low notes when set to "Geetar".