Eisenhower
Eisenhower, as General of the Allied Armies, had plenty of executive experience and public exposure. In addition, he built his political resume by acting as President of Columbia University and did not run for President until 1952. This was years after he became a household name.
West has served a term in the House and is just beginning his public service career and is in no way a household name. With continued success and another 4 to 8 yrs of seasoning he might be ready for the big leagues.
Victorious generals are an exception to the professional politicians who have been our Presidents. Hoover and Taft are as well if you want to discuss them.
However, all have had significant time on the public stage.