Hi Daryl,  According to the "Pierce Piano Atlas 10th Edition",
Hamilton Piano #16566 was made in the first year of manufacture:
1900.  Serial numbers in 1900 started with #16000, and 1901 started
with 20000.

  Hamilton Piano Co., Est. 1889.
  Address 1801 Gilbert, Cincinnati, Ohio.
  Previously at Chicago Heights, Illinois.
  Made by Baldwin Piano & Organ Co.

  Baldwin Piano & Organ Co., Est. 1862, makers of Baldwin, Hamilton,
  Howard, Ellington, Valley Gem, St. Regis, Schroeder, Franke,
  Acrosonic, Kremlin, Modelle, Monarch, Sargent, Winton, Chickering and
  Wurlitzer.  Current address, 422 Wards Corner Road, Loveland Ohio.

The Atlas lists serial numbers from 1890 to 1995.  Presto Buyers Guide,
1926, lists the following:

  Hamilton Chicago factories are very large and form an important
  part of the output controlled by the Baldwin Piano Co.  Hamilton
  pianos, comprising upright, grand and reproducing pianos, school
  special pianos as Manualo player-pianos are instruments of decided
  merit and are very popular throughout the United States.

  They formed a part of the Baldwin exhibit at the Paris Exposition
  of 1900, and attracted much attention there.  Many agencies were
  established for their representation in foreign lands.  The silver
  metal was awarded the Hamilton pianos by the exhibition in
  Melbourne, 1903.

  The Hamilton Manualo is a player-piano of extraordinary merit.  It
  combines the qualities of the Hamilton pianos with a scientifically
  constructed player mechanism.  It is absolutely sure to give lasting
  satisfaction.

Jack M. Conway, Los Angeles, California, USA