Several nearby individual low column density interstellar cloudlets have been identified previously on the basis of kinematical features evident in high-resolution Ca⁺ observations near the Sun. One of these cloudlets, the "Apex Cloud" (AC), is within 5 pc of the Sun in the solar apex direction. The question of which interstellar cloud will constitute the next Galactic environment of the Sun can, in principle, be determined from cloudlet velocities. The interstellar absorption lines toward α Cen (the nearest star) are consistent within measurement uncertainties with the projected "G" cloud (GC) and AC velocities, and also with the velocity of the cloud inside of the solar system (the local interstellar cloud [LIC]), provided a small velocity gradient is present in the LIC. The high GC column density toward α Oph compared to α Aql suggests that α Aql may be embedded in the GC so that the AC would be closer to the Sun than the GC. This scenario favors the AC as the next cloud to be encountered by the Sun, and the AC would have a supersonic velocity with respect to the LIC. The weak feature at the AC velocity toward 36 Oph suggests that the AC cloud is either patchy or does not extend to this direction. Alternatively, if the GC is the cloud that is foreground to α Cen, the similar values for N(H⁰) in the GC components toward α Cen and 36 Oph indicate this cloud is entirely contained within the nearest ~1.3 pc, and the Ca⁺ GC data toward α Oph would then imply a cloud volume density of ~5 cm^-3, with dramatic consequences for the heliosphere in the near future.