Efficiency of a root finding algorithm
What you describe is a reasonable approach.
Your first step is often called root isolation, and your second step is (in a sense) root polishing. You can Google these terms to learn more.
In your second step, the end-points of your interval bracket a root. It’s a good idea to use a polishing algorithm that keeps the root bracketed. The standard Newton-Raphson algorithm does not preserve bracketing, but it’s easy to adapt it so that it does. There’s a decent implementation in the Numerical Recipes book.