Other than in yod'ei ha'ittim, so far as I can tell the word eis appears in the Megillah three times:
4:14 - bo'eis ha'zos (Mordechai speaking to Esther).
5:13 - b'chol eis (Haman to his household and Zeresh).
8:9 - bo'eis ha'hee (the 23rd of Sivan, when the counter-decree was written).
According to the Malbim (HaCarmel), eis connotes a natural point or span of time that is manifest in nature - as opposed to a point in time determined by dos (he uses the word!) and human convention. An eis can be long: geshamim b'ittam - or short: l'eis erev.
Zman, on the other hand, is not part of an ebb and flow in the tides of natural phenomena. Although the Malbim does not state this in the cited passage, it is clear that zman and hazmanah are related.
Yom is also a set time, defined by the phenomena of night and day. There can be several ittim in a day and, conversely, many days part of a larger eis. Thus, v'al yavo b'chol eis el ha'kodesh, says the Malbim, means not even at all ittim during the day of Yom Kippur.
To be Hirschian, the shoresh of eis, עתת is related to the shoresh of os, אתת. They are both significations, v'duk.
Moadim in the Torah are connected with ittim - seasons. Yomim Tovim d'Rabbanan are connected with zmanim - times at which events occurred independent of seasons of harvest, first fruits and gathering. Avodah Zarah and soothsaying are concerned with seasons and auspicious phenomena that indicates a fortuitous point or span of time. Atheism denies any such significance. There is only time.
If you believe in coincidences, a mikreh is meaningless. It also has nothing to do with zman, other than that it takes place within time. But if you believe in mazal and/or hashgachah, a mikreh may very well indicate the onset of an eis.
One more hakdamah: In the Moreh 1:73, the Rambam notes the error of the Medabrim (the Mutakallemin) who see time as comprised of an infinite number of "time-atoms" (Friedlander's translation). The Rambam holds that time is not comprised of "time-atoms" but is an indivisible unity...