Many former Republicans are now Libertarians. So many Republicans harbor Libertarian beliefs that there is a general term for them . . . Libertarian Republicans.
From Wikipedia . . .
Sometimes the terms republitarian or liberty Republican are used as well. Libertarian Republicans' views are similar to Libertarian Party members, but differ in regard to the strategy used to implement libertarian policies and on some issues of civil liberties. Libertarian Republicans represent a political faction within the Republican Party.
From The Libertarian Republican website . . .
What's a Libertarian Republican?
Libertarian Republicans are libertarians who believe in working within the Republican Party to advance the agenda of liberty. Many libertarian Republicans are former member of the Libertarian Party who have left to join the GOP. Others keep their current affiliation with the Libertarian Party while still participating in Republican politics, as well. Still others are lifelong Republicans who have always felt a kinship to the libertarian movement, and libertarian beliefs. Most libertarian Republicans are active to some degree with the Republican Liberty Caucus, which represents the libertarian wing of the GOP.
From the Illinois Libertarian Party website FAQ
Are Libertarians more like Democrats or Republicans?
Libertarians take the sensible center position between the left and the right.
Many argue that there are two conservative parties--Republicans and Libertarians. Check it out.
Libertarian party is an ideal choice for the college conservative