First observation of new effects was carried out using the set-up created for searching for a neutron electric dipole moment (EDM) by a crystal-diffraction method. For the first time the neutron dynamical Laue diffraction for the Bragg angles close to a right angle (up to 87°) was studied, using the direct diffraction beam and a thick (∼ 3.5-cm) crystal. The effect of an essential time delay of diffracting neutrons inside the crystal for Bragg angles close to 90° was experimentally observed, using a time-of-flight method. The phenomenon of neutron-beam depolarization was first experimentally observed for the case of Laue diffraction in a noncentrosymmetric α-quartz crystal. It is experimentally proved that the interplanar electric field, affecting a neutron in a crystal, maintains its value up to Bragg angles equal to 87°. These results confirm the opportunity to increase by more than an order of magnitude the sensitivity of the method to the neutron EDM, using the diffraction angles close to 90°, and give a real prospect to exceed the sensitivity of the magnetic resonance method, using ultra-cold neutrons.