The praefix ex/e/ec- has an 'original' meaning of "out, away" (emitto, evanesco), which also developed into a more general idea of privation or even negation, like "un-" (exonero).
Another later sense is "throughout, to the end", which additionally developed into "thoroughly, completely" (evinco, enarro) in turn.
I suspect eviro "to unman" has the sense of privation, like emasculo "to unman"; but it is not entirely clear in which sense the praefix was used to form effemino.
None of the dictionaries I consulted seem to provide conclusive evidence. I think it may have been the sense of "thoroughly, completely", resulting in "to become utterly feminine".
Alternatively, it may have been formed by some sort of inverted (sloppy?) analogy with eviro and emasculo. It would be an odd analogy, almost the opposite of those verbs, but etymology works in strange ways.
Lewis & Short suggest ec- in effemino has the sense "out of (a former nature)", but I find their reasoning somewhat inconsistent:
Primarily and most freq. of place, out or forth: exeo, elabor, educo, evado, etc.; and in an upward direction: emineo, effervesco, effero, erigo, exsurgo, exsulto, extollo, everto, etc.—
Hence also, trop., out of (a former nature), as in effeminare, qs. to change out of his own nature into that of a woman: effero, are, to render wild; thus ex comes to denote privation or negation, Engl. un-: exanimare, excusare, enodare, exonerare, effrenare, egelidus, I., elinguis, elumbis, etc.—
While effero (efferare), "to make wild", works in the same direction as effemino, their other examples do not: exanimare does not mean "to deprive someone out of his nature into an anima", but rather "to deprive someone of his anima". The root word is the thing one is deprived of, not the thing one is turned into.
Both effero and effemino could be explained based on the sense "thoroughly, completely".
The Oxford Latin Dictionary doesn't tell us much about any of this. De Vaan doesn't mention effemino nor effero, nor yet this specific sense of ex/e/ec, under neither femina nor ferus, nor yet under ex/e/ec, so he probably doesn't have any specific information either.