This is gonna sound messed up but this is what you need to get an em6
Put in
C#m7 b5/E
6 chords are actually inverted chords but can be used in this way, but it does mean that the theory is a bit confusing to get them diatonically, especially as the c# in an em6 isn’t in the e minor scale.
Using the key of D Major, or B Minor instead of E minor, will mean that c# is in key.