We could find the answer by trial and error -- testing each candidate $t$ to see if $t\cdot (t+2)\equiv 1\pmod{23}$. However, here is another way:

We can easily see that $4\cdot 6=24\equiv 1\pmod{23}$, so $4$ fulfills the main requirement that its inverse is $2$ more than it. Unfortunately, $4$ isn't odd. But we also have \begin{align*}
(-4)\cdot (-6) &= 4\cdot 6 \\
&\equiv 1\pmod{23},
\end{align*} so $-4$ and $-6$ are each other's inverses $\pmod{23}$. Since $-4\equiv 19\pmod{23}$ and $-6\equiv 17\pmod{23}$, the answer $t=\boxed{17}$ satisfies the requirements of the problem.

(We can even check that $17\cdot 19 = 323 = 14\cdot 23 + 1$.)
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Find the unique $\textbf{odd}$ integer $t$ such that $0<t<23$ and $t+2$ is the inverse of $t$ modulo $23$.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
17
Find the unique $\textbf{odd}$ integer $t$ such that $0<t<23$ and $t+2$ is the inverse of $t$ modulo $23$.
test