Mary chose an even $4$-digit number $n$. She wrote down all the divisors of $n$ in increasing order from left to right: $1,2,...,\tfrac{n}{2},n$. At some moment Mary wrote $323$ as a divisor of $n$. What is the smallest possible value of the next divisor written to the right of $323$?

$\textbf{(A) } 324 \qquad \textbf{(B) } 330 \qquad \textbf{(C) } 340 \qquad \textbf{(D) } 361 \qquad \textbf{(E) } 646$
1. Let the next divisor be \( k \). We need to find the smallest possible value of \( k \) such that \( k > 323 \) and \( k \) is a divisor of \( n \).
2. Note that \( 323 = 17 \times 19 \). Therefore, \( k \) must be a multiple of either 17 or 19 to ensure that \( n \) remains a 4-digit number.
3. The smallest multiple of 17 greater than 323 is \( 17 \times 20 = 340 \).
4. The smallest multiple of 19 greater than 323 is \( 19 \times 18 = 342 \).
5. We need to check if \( k = 340 \) works. If \( k = 340 \), then \( n \) must be a multiple of both 323 and 340. The least common multiple (LCM) of 323 and 340 is:
   \[
   \text{lcm}(323, 340) = \text{lcm}(17 \times 19, 2^2 \times 5 \times 17) = 17 \times 19 \times 2^2 \times 5 = 6460
   \]
6. Since 6460 is a 4-digit number, it is a valid candidate for \( n \).
7. Therefore, the smallest possible value of the next divisor written to the right of 323 is \( 340 \).

The final answer is \(\boxed{340}\).