What you clearly are unfamiliar with is that you are freer crossing international borders within Schengen than to/from the UK which has a controlled border. For the purposes of your question Eire is part of the UK. Once you are out of the UK and into Schengen there are no border controls until you leave Schengen and you need not worry regarding your movements. If you had the inclination, you could take a train (or flight/ferry or drive) from Amsterdam to Iceland, northern Norway/Sween/Finland or southern Portugal/Spain/France/Italy/Greece and get back to your flight from Amsterdam without ever having to go through border checks or submit the CIA passport/contact/personal data required whenever you fly to/from the UK (I am assuming Eire as well, am unfamiliar). Enjoy the freedom once you are there.