A budget of up to £600 for a hardtail frame gives you a pretty good range of choice.
If you’re looking at transferring parts, the things you need to pay particular attention to are:
Wheel size.
Steerer type (almost certainly tapered now but worth making sure your existing fork isn’t straight 1 1/8)
Head tube length vs Steerer length of your existing forks. 29ers usually have quite short headtubes to keep front ends low.
Seat tube diameter and whether there are obstructions in the seat tube that will stop you dropping your post if not using a dropper.
Bottom bracket standard. There are many.
With these in mind, decide what sort of use you intend and look at frames with suitable geometry.
Don’t use regular grease on carbon parts, but a tube of carbon Grip stuff and It’s highly recommended you use a torque wrench for carbon.