"""""Is there a possibility or a trick to extend an arc-shaped curve while maintaining its overall shape?""""

So the extend will just make a tangent offshoot...

A "trick" would be to know and create a line that is at the end of, and is as your your desired result. (Position-direction)

Then you can use blend, which will allow tangency and curvature to be maintained over distance

With history enabled, you can interactively drag and rotate to get the extended shape you want...