[Agrippa]

I would mainly change to laws for that matter to change things a little bit as a first step:
a) If a woman lies about the father thats a serious crime and will be punished.
b) If a woman cheats in the marriage she will lose all her rights (financial etc.) and in extreme cases the parental care and punishment either - of course she can get a divorce, but she must first do it then she can have another relationship.

Now marriage doesnt mean to much any more, it made sense once it was relative social security for the female and sexual and personal security for the male + a production and life unity. Thats lost now and marriage doesnt make a big difference at all.
Furthermore females acting amoralic and destructive have all rights on their side, these two steps alone would change the direction, though more measures would be necessary.

And it would be a rather unpleasant experience to have sex with a women which is pregnant from another guy, not really acceptable to me, if she would have lied either, much worse...

I can just agree with Scoob.