Selecting different letters like "ä" and "õ" on IPhone SE I have to press and hold the the default letter "a" for example which then opens another list of modified letters which I can then choose If I slide my finger without lifting it up to the "ä".

I find it inconvenient, is there a way to set it that you when you press and hold the default letter and lift up your finger that the alternative letter menu still stays up?

Or any other  suggestions which you think might be more preferrable to me?
There is no way to make the Character Picker stay up.

The alternative is to use a keyboard which has the letters you need already directly on it.  Apple's Estonian doesn't do that, so have a look at the Keyman app or other 3rd party keyboards from the App Store for this language.