Sources
Census Office. Ninth Census of the United States (1870). Volume I, pages 11-74. Table II: "Population by Counties—1790–1870".

Table II: "Population by Counties—1870–1870" is organized in sections named for states and territories, in alphabetical order. Within each section the Counties (so-called by the column label) are listed generally in alphabetical order. There are 2356 rows within 47 sections.

Here are the numbers of "Counties" by section with explanation of all departures from alphabetical order.

zero recs rows null State or Terr departures from alphabetical "Counties"
0 65 68 3 Alabama
4 0 Arizona T
0 61 62 1 Arkansas
0 50 50 0 California
0 21 21 0 Colorado T
0 8 8 0 Connecticut
15 0 Dakota T 14 + "Unorganized ..."
0 3 3 0 Delaware
4 1 District of C Georgetown..., Washington..., Remainder..., Alexandria...
0 39 43 4 Florida
0 132 133 1 Georgia
9 0 Idaho T
0 102 102 0 Illinois
0 92 93 1 Indiana
1 99 100 2 Iowa
4 68 72 8 Kansas
0 115 115 0 Kentucky
0 53 53 0 Louisiana
0 16 16 0 Maine
1 23 22 0 Maryland
0 14 14 0 Massachusetts
5 76 80 9 Michigan
5 76 81 10 Minnesota
0 65 66 1 Mississippi
0 114 117 3 Missouri
11 0 Montana T
5 57 36 5 Nebraska 32 + two "Unorganized ..." + two "... Indian Reservation"
3 17 17 3 Nevada
0 10 10 0 New Hampshire
0 21 21 0 New Jersey
14 1 New Mexico T
0 60 60 0 New York
0 90 90 0 North Carolina
0 88 88 0 Ohio
0 22 25 3 Oregon
0 66 66 0 Pennsylvania
0 5 5 0 Rhode Island
0 31 32 1 South Carolina
0 85 85 0 Tennessee
18 159 162 21 Texas
27 6 Utah T
0 14 14 0 Vermont
0 99 99 0 Virginia
24 2 Washington T 23 + "The Disputed Islands"
0 53 53 0 West Virginia
0 58 61 3 Wisconsin
5 0 Wyoming T
42 2227 2356 89 Total
42 2206 2222 79 Subt, States
0 21 134 10 Subt, Terr + DC
zero recs rows null State or Terr departures from alphabetical "Counties"

There are 47 sections with names that match 45 modern states (the fifty less Alaska, Hawaii, North Dakota, Oklahoma, and South Dakota) plus Dakota Territory and the District of Columbia. Every one includes multiple rows for so-called counties. Bold T marks nine territories in contrast to the 37 states and DC.