It is a theorem of Stoimenow that there exist unknotting number one knots with minimal crossing diagrams of unknotting number greater than one. Two such examples are $14_{36750}$ and $14_{36760}$. See Figure 9 in the reference:

- A. Stoimenow. Some examples related to 4-genera, unknotting numbers and knot polynomials. J. London Math. Soc. (2), 63(2):487–500, 2001.

There is a related result of of Bleiler and Nakanishi (independently) that the knot $10_8$ admits a 14-crossing diagram of unknotting number three --- yet all minimal crossing diagrams have unknotting number four!

Steven A. Bleiler. A note on unknotting number. Math. Proc. Cam-
bridge Philos. Soc., 96(3):469–471, 1984.

Yasutaka Nakanishi. Unknotting numbers and knot diagrams with the
minimum crossings. Math. Sem. Notes Kobe Univ., 11(2):257–258, 1983.

There are is a discussion of these examples and some nice figures in Staron's thesis.