  We explore the connections between various coordinate systems associated with
observers moving inwardly along radial geodesics in the Schwarzschild geometry.
Painlev\'e-Gullstrand (PG) time is adapted to freely falling observers dropped
from rest from infinity; Lake-Martel-Poisson (LMP) time coordinates are adapted
to observers who start at infinity with non-zero initial inward velocity;
Gautreau-Hoffmann (GH) time coordinates are adapted to observers dropped from
rest from a finite distance from the black hole horizon. We construct from
these an LMP family and a proper-time family of time coordinates, the
intersection of which is PG time. We demonstrate that these coordinate families
are distinct, but related, one-parameter generalizations of PG time, and show
linkage to Lema\^itre coordinates as well.
