  The $^{12}$C($\alpha,\gamma$)$^{16}$O reaction, an important component of
stellar helium burning, has a key role in nuclear astrophysics. It has direct
impact on the evolution and final state of massive stars and also influences
the elemental abundances resulting from nucleosynthesis in such stars.
Providing a reliable estimate for the energy dependence of this reaction at
stellar helium burning temperatures has been a longstanding and important goal.
In this work, we study the role of potential new measurements of the reaction,
$^{16}$O($e,e'\alpha$)$^{12}$C reaction, in reducing the overall uncertainty. A
multilevel $R$-matrix analysis is used to make extrapolations of the
astrophysical S factor for the $^{12}$C($\alpha,\gamma$)$^{16}$O reaction to
the stellar energy of 300 keV. The statistical precision of the $S$-factor
extrapolation is determined by performing multiple fits to existing $E1$ and
$E2$ ground state capture data, including the impact of possible future
measurements of the $^{16}$O($e,e'\alpha$)$^{12}$C reaction. In particular, we
consider a proposed MIT experiment that would make use of a high-intensity
low-energy electron beam that impinges on a windowless oxygen gas target as a
means to determine the total $E1$ and $E2$ cross sections for this reaction.
