  We report stronger constraints on the parameters of Yukawa-type corrections
to Newtonian gravity from measurements of the lateral Casimir force between
sinusoidally corrugated surfaces of a sphere and a plate. In the interaction
range from 1.6 to 14 nm the strengthening of previously known high confidence
constraints up to a factor of $2.4\times 10^7$ is achieved using these
measurements. It is shown that the replacement of a plane plate with a
corrugated one in the measurements of the normal Casimir force by means of an
atomic force microscope would result in the strengthening of respective high
confidence constraints on the Yukawa-type interaction by a factor of $1.1\times
10^{12}$. The use of a corrugated plate instead of a plane plate in the
experiment by means of a micromachined oscillator also leads to strengthening
of the obtained constraints. We further obtain constraints on the parameters of
Yukawa-type interaction from the data of experiments measuring the gradient of
the Casimir pressure between two parallel plates and the gradient of the
Casimir-Polder force between an atom and a plate. The obtained results are
compared with the previously known constraints. The possibilities of how to
further strengthen the constraints on non-Newtonian gravity are discussed.
