  Searches for planets via gravitational lensing have focused on cases in which
the projected separation, a, between planet and star is comparable to the
Einstein radius, R_E. This paper considers smaller orbital separations and
demonstrates that evidence of close-orbit planets can be found in the
low-magnification portion of the light curves generated by the central star. We
develop a protocol to discover hot Jupiters as well as Neptune-mass and
Earth-mass planets in the stellar habitable zone. When planets are not
discovered, our method can be used to quantify the probability that the lens
star does not have planets within specified ranges of the orbital separation
and mass ratio. Nearby close-orbit planets discovered by lensing can be subject
to follow-up observations to study the newly-discovered planets or to discover
other planets orbiting the same star. Careful study of the low-magnification
portions of lensing light curves should produce, in addition to the discoveries
of close-orbit planets, definite detections of wide-orbit planets through the
discovery of "repeating" lensing events. We show that events exhibiting
extremely high magnification can effectively be probed for planets in close,
intermediate, and wide distance regimes simply by adding several-time-per-night
monitoring in the low-magnification wings, possibly leading to gravitational
lensing discoveries of multiple planets occupying a broad range of orbits, from
close to wide, in a single planetary system.
