I don't know exactly what you mean by "stand by operation". You mean sleep? 20mA20 mA is pretty high for sleep. I'd expect < 100uAless than 100 uA in general.
I'd go with SD Cardcard storage. I've designed plenty of "low power systems" and the reality is that most people go to the trouble of going all the way to optimize for low power when in fact it doesn't make that big of an impact (Itit makes for nice marketing, but marketing doesn't always makes real world-world sense). There are many other factorfactors that are more important to logging:
Accessibility - An SD Cardcard is much easier and faster to read from and access (just take it out, pop in an SD Cardcard reader and it's done), and it's faster to do. EEPROM reading requires more equipment. An SD Cardcard can be accessed using a file system so it's intuitive
Capacity - You won't find anything better than a couple of gigsgigabytes of storage. It's amazing how fast this goes once you startsstart logging.
Ease of Implementation - Many commercial and open source appsapplications have used FATFSFatFs which you can get for many microcontrollers. It will work nicely. Just open a txt file and save the numbers, one after the other.
An SD Cardcard isn't that power hungry. Some have power down-down modes after 5ms5 ms that will help.
I can't find a reason not to use an SD Cardcard.