I don't know exactly what you mean by "stand by operation". You mean sleep? 20mA is pretty high for sleep. I'd expect < 100uA in general.

I'd go with SD Card 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 (It makes nice marketing, but marketing doesn't always makes real world sense). There are many other factor that are more important to logging:

1) Accessibility - SD Card is much easier and faster to read from and access (just take it out, pop in SD Card reader and it's done), and it's faster to do. EEPROM reading requires more equipment. SD Card can be accessed using file system so it's intuitive

2) Capacity - You won't find anything better than a couple of gigs of storage. It's amazing how fast this goes once you starts logging

3) Ease of Implementation - Many commercial and open source apps have used FATFs which you can get for many microcontrollers. It will work nicely. Just open a txt file and save the numbers, one after the other.

SD Card isn't that power hungry. Some have power down modes after 5ms that will help.

I can't find a reason not to use SD Card.