I don't know whether you're up for suggestions, but here's one:

\documentclass{article}
\newcounter{hintcntr}
%\renewcommand{\thehintcntr}{\arabic{hintcntr}}
\makeatletter
\newenvironment{hints}
{% \begin{hints}
\setcounter{hintcntr}{0}% Restart numbering
\renewcommand{\item}{\stepcounter{hintcntr}\@ifstar\@itemstar\@itemnostar}
\def\@itemstar{\ignorespaces}% \item*
\def\@itemnostar{\ignorespaces\thehintcntr.~}% \item
\par%
[~Hint:%
}
{\unskip~]}% \end{hints}
\makeatother
\begin{document}
\begin{enumerate}
\item First item
\item Second item
\item Third item
\item Last item
\end{enumerate}
\begin{hints}
\item Hint for first item in list
\item Hint for second item in list
\item*% no hint here
\item Hint for last item in list
\end{hints}
\end{document}
The hints environment sets a regular paragraph with \item printing the next hint, while \item* just steps the hint counter without printing anything.
Here's an implementation of the hints environment that matches in output what you're currently requesting:

\newcounter{hintcntr}
%\renewcommand{\thehintcntr}{\arabic{hintcntr}}
\makeatletter
\newenvironment{hints}
{% \begin{hints}
\setcounter{hintcntr}{0}% Restart numbering
\def\newpar{\def\newpar{\par}}% http://tex.stackexchange.com/a/89187/5764
\renewcommand{\item}{\stepcounter{hintcntr}\@ifstar\@itemstar\@itemnostar}
\def\@itemstar{\ignorespaces}% \item*
\def\@itemnostar{\ignorespaces\newpar\noindent\thehintcntr.~}% \item
\par\noindent%
[~Hint:%
}
{\unskip~]}% \end{hints}
\makeatother
inline' for packageenumitem', Missing number, treated as zero\item. What are these? – Soumitra Sen Mar 17 '14 at 22:54enumitem. Please update the package. – Gonzalo Medina Mar 17 '14 at 22:59