You told us several times that you need more precision but never what your precision limit is.
You have two possibilities to get more precision:
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Get an Arduino with a crystal as clock source and not a resonator (as the UNO has). This increases the precision by a factor of around 1000.
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Take the advice from dannable and buy a DS3231. It's does not only have temperature compensated RTC but also two programmable interrupts. You can program your next log action and it will be timed very accurately (although you don't get millisecond resolution).