To find the coefficient of the second term in the expansion of \((a+3)^7\) using the Binomial Theorem, we start by recognizing the general form of the Binomial Theorem:
\[
(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k
\]
In this case, we can let \(x = a\), \(y = 3\), and \(n = 7\). The expansion of \((a + 3)^7\) will therefore be:
\[
(a + 3)^7 = \sum_{k=0}^{7} \binom{7}{k} a^{7-k} \cdot 3^k
\]
We are interested in the coefficient of the second term in this expansion. The second term corresponds to \(k = 1\) (noting that the first term corresponds to \(k = 0\)).
To compute the coefficient of the second term, we calculate:
\[
\text{Coefficient} = \binom{7}{1} \cdot 3^1
\]
We first evaluate \(\binom{7}{1}\):
\[
\binom{7}{1} = 7
\]
Next, we simplify \(3^1\):
\[
3^1 = 3
\]
Now, we multiply these results to find the coefficient of the second term:
\[
\text{Coefficient} = 7 \cdot 3 = 21
\]
Thus, the coefficient of the second term in the expanded form of \((a + 3)^7\) is \(\boxed{21}\).