ref to subsection number only

Related to your MWE you can use the following:

\documentclass{article}
\makeatletter
\def\@firstoftwo@second#1#2{%
  \def\temp##1.##2\@nil{##2}%
   \temp#1\@nil}
\newcommand\sref[1]{%
   \expandafter\@setref\csname r@#1\endcsname\@firstoftwo@second{#1}%
}

\makeatother
\begin{document}
\section{This is my section}
For further information see
\begin{tabular}{ll}
 \verb+\sref{subsecone}+   & \sref{subsecone} \\
 \verb+\sref{subsectwo}+   & \sref{subsectwo} \\
 \verb+\sref{subsecthree}+ & \sref{subsecthree} \\
\end{tabular}  within this section.
\subsection{One}\label{subsecone}
\subsection{Two}\label{subsectwo}
\subsection{Three}\label{subsecthree}
\end{document}

---

Some explanation.

The standard reference are saved in the aux-file in the following form:

\newlabel{subsectwo}{{1.2}{1}}

You can see that \newlabel is a command with two mandatory argument. The first argument is the label name and the second argument has two other groups inside. The first group is the numbering scheme and the second group shows the page.

So you want to extract the first group of the second mandatory argument and then you want to extract the second part of the number.

With the command \@firstoftwo@second you take the first group of the second argument of \newlabel and due to it's defintion you extract your requested number.