I started hearing this word in various anime: it's starting to seem quite common, but I'm not sure about its meaning.
Jisho lists it as "sincerely; repeatedly", but it doesn't seem to be it's whole meaning, if at all; unfortunately I wasn't able to find the examples where I heard it in the past, but I took a couple from Jisho and one from the last time I heard it:
We are counting on you from tomorrow
Give him my best regards
Please take good care of yourself
Please be careful
It seems to me in those examples it doesn't really translate as per Jisho: if it's translated at all (in the first example it doesn't seem to be translated), it in "best regards" and "good care".
I tried Weblio, but as far as I can understand it says it means "To repeat many times; used when asking sincerely", which seems more or less in line with Jisho and again doesn't seem to really fit.
Am I right in thinking that 「くれぐれも」 is best understood as an intensifier, a word used to put emphasis on what is being said, rather than having specific meanings like those given by Jisho? If I'm right, 「明日からくれぐれも頼んだよ」 wold mean something like "We are really counting on you from tomorrow", "We are counting on you from tomorrow [and we are expecting results]".
(Also, I'm just realizing this while finishing this question, I'm not sure why with 「明日」 is used 「頼んだ」, a past form.)