  We consider the CANNEX (Casimir And Non-Newtonian force EXperiment) test of
the quantum vacuum intended for measuring the gradient of the Casimir pressure
between two flat parallel plates at large separations and constraining
parameters of the chameleon model of dark energy in cosmology. A modification
of the measurement scheme is proposed that allows simultaneous measurements of
both the Casimir pressure and its gradient in one experiment. It is shown that
with several improvements the CANNEX test will be capable to strengthen the
constraints on the parameters of the Yukawa-type interaction by up to an order
of magnitude over a wide interaction range. The constraints on the coupling
constants between nucleons and axion-like particles, which are considered as
the most probable constituents of dark matter, could also be strengthened over
a region of axion masses from 1 to 100 meV.
