[Tex/LaTex] Subscript after \underline in math mode

subscriptsunderline

I would like to put a subscript after a character with multiple underlines in the math mode but ignoring the underlines. How can I do it?

Consider this picture:
vertical subscript positions
I would like the subscript to have the vertical position of the element with respect to the character. Here is the code the produced the picture:

\documentclass[border=5mm]{standalone}
\usepackage{ulem,amsmath,bm}
\begin{document}
\(C_{\text{c}}, 
\underline{\bm{C}}_{\text{c}}, 
\underline{\underline{\bm{C}}}_{\text{c}}, 
\uuline{\bm{C}}_{\text{c}}\)
\end{document}

Edit: Would be superb if the underline was limited to the width of the character.

Best Answer

Thanks to this answer, the following satisfied my requirements, including additional:

\newcommand{\ubar}[1]{\mkern 1.5mu\underline{\mkern-1.5mu#1\mkern-1.5mu}\mkern 1.5mu}
\newcommand{\uubar}[1]{\ubar{\ubar{#1}}}