jmc

Using long division, we find that $\frac{3}{26}$ can be expressed as a repeating decimal $0.1\overline{153846}$.

After the first digit, there is a six-digit repeating block. We want to find the $99$th digit after the first digit. The remainder when $99$ is divided by $6$ is $3$. Therefore, the $100$th digit is the third digit in the repeating block, which is $\boxed{3}$.