Another way of adjusting the spacing aft …

Another way of adjusting the spacing after line with stretched table
Use bookends package and \toprule, \midrule, and \bottomrule instead of \hline.

In premeble

\usepackage{booktabs}

In context

{table}[h]
\begin{tabular*}{\linewidth}{@{\extracolsep{\fill}} ccr}
\toprule
Site pattern & supported $T_i$ & count \\
\midrule
xxx & $T_0$ & 4 \\ 
xxy & $T_1$ & 3 \\ 
\bottomrule
\end{tabular*}
\caption{Table1}
\end{table}