It's a couple of years since I had to transit at STN but at that time if you didn't already have a boarding card for your onward flight you had to leave bagage claim and check in at the public check-in desks.

If you already have a boarding pass (for an international connection) you can go from baggage claim back into the departure lounge.

Either way, you still have to take the train-shuttle to the main terminal building and then back again to your gate (min round trip time of 15mins or so).

Since you're getting off one KLM flight and on to another (hand lungage only, right?) you may be able to check in at the gate and not travel to the main terminal.

If you have to check in at the normal check-in desks in the main terminal and your are short of time then ask for a last boaring card and/or a green Fast Track sticker (also available for SW/RW).

Whew! Hope this helps!

- Dambus