New core designs do in fact become available as you research the required techs but you have to research *all* techs required by the design before they are available. Don't forget you might need life support and miniaturization techs as well as the obvious weapons, defense, engine and hull techs.
However in general the core ships suck and you really should just get into the habit of designing your own ships. They don't have to look pretty they just have to work and you're far better off designing your own ships that use the best of whatever techs you currently have instead of not being able to build some predefined ship just because your life support or sensors aren't up to snuff.
You can just pick a hull and use the place button if you don't care how the ship looks. It will probably end up looking lopsided but who cares? And if you do care like me then it really doesn't take all that much to make the ship look symmetrical with weapons in the front and engines in the back.
You really are limiting yourself by being dependent on the core designs because all in all they aren't really all that useful. I usually upgrade my miner to another colony ship and once both colony ships are gone I "obsolete" all the core designs I start with and never use another core design ever. I'm pretty sure that's what most folks do as well.