  In this article, we study the logarithmic Mahler measure of the one-parameter
family \[Q_\alpha=y^2+(x^2-\alpha x)y+x,\] denoted by $m(Q_\alpha)$. The zero
loci of $Q_\alpha$ generically define elliptic curves $E_\alpha$ which are
$3$-isogenous to the family of Hessian elliptic curves. We are particularly
interested in the case $\alpha\in (-1,3)$, which has not been considered in the
literature due to certain subtleties. For $\alpha$ in this interval, we
establish a hypergeometric formula for the (modified) Mahler measure of
$Q_\alpha$, denoted by $\tilde{n}(\alpha).$ This formula coincides, up to a
constant factor, with the known formula for $m(Q_\alpha)$ with $|\alpha|$
sufficiently large. In addition, we verify numerically that if $\alpha^3$ is an
integer, then $\tilde{n}(\alpha)$ is a rational multiple of $L'(E_\alpha,0)$. A
proof of this identity for $\alpha=2$, which is corresponding to an elliptic
curve of conductor $19$, is given.
