  It is conjectured that all perturbative approaches to quantum electrodynamics
(QED) break down in the collision of a high-energy electron beam with an
intense laser, when the laser fields are boosted to `supercritical' strengths
far greater than the critical field of QED. As field strengths increase toward
this regime, cascades of photon emission and electron-positron pair creation
are expected, as well as the onset of substantial radiative corrections. Here
we identify the important role played by the collision angle in mitigating
energy losses to photon emission that would otherwise prevent the electrons
reaching the supercritical regime. We show that a collision between an electron
beam with energy in the tens of GeV and a laser pulse of intensity
$10^{24}~\text{W}\text{cm}^{-2}$ at oblique, or even normal, incidence is a
viable platform for studying the breakdown of perturbative strong-field QED.
Our results have implications for the design of near-term experiments as they
predict that certain quantum effects are enhanced at oblique incidence.
