Let the amount of money Conner has be represented as \( C \).
According to the problem, if Conner had another $6, he would have three times as much money as Rei. Since Rei has $18, we can set up the equation:
\[ C + 6 = 3 \times 18 \]
Calculating \( 3 \times 18 \):
\[ 3 \times 18 = 54 \]
Now we can substitute this into our equation:
\[ C + 6 = 54 \]
To find \( C \), we subtract 6 from both sides:
\[ C = 54 - 6 \]
Calculating that gives:
\[ C = 48 \]
Therefore, Conner has \(\boxed{48}\) dollars.