There is an extensive article I've been working on in the Hearth Wiki that addresses the entire subject of hearth pad design and construction. It is
HERE! I would suggest that you start by reading the article, and then asking any questions you have left...
In short, building an extension is basically the same as building a complete hearth pad, the same principles apply. You need to determine the size (in this case of just the extension) and R-value requirements, then choose appropriate materials and design to fit the requirements.
IMPORTANT NOTE - The Code reqirements spelled out in the NFPA 211 standards have just changed! You now must have at least
18 inches in front of your loading doors, not the 16" specified for the US in many manuals - make sure your design gives you this or you may have problems with your inspection.
I would verify that your existing hearth meets the R-value requirements for the stove you are planning to install, and build your extension to the same R-value as the existing hearth.
One thing I haven't put in the article yet, but need to, is that if you are building a split level hearth, the stove MUST be on the highest part of it - you can't build an extension that sticks up higher than the area the stove is sitting on. (It is OK if the extension is lower)
i was thinking about either making my own pad? or cuting out the carpet for 12” and installing some ceramic floor tile in front of the existing hearth.
I think cutting the carpet gives better results. Note that much depends on your installation, a low R-value stove, or one that just says "Non-combustible floor protection" will probably let you get away with just a layer of tiles instead of the carpet. If you need more than that, it is possible to also cut out a section of the sub-floor (and if really needed, with more difficulty) the decking under it, and build back up with some combination of insulating materials to reach or exceed the required R-value, and still come up with a flush installation. (See the above article for details)
A few other things that will help us give better advice...
What does the current setup look like? - Photos are helpful!
What is the R-value of the existing hearth?
Is the existing hearth flush with the floor, or raised? If raised, how high is it?
What kind of stove are you planning to install? What does the manual say it requires for R-Value on the floor protection?
Gooserider