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She can only travel to states with where the trailing letter of the current state's capital matches the first letter of destination state capital.

She can only travel to states with where the trailing letter of the current state's capital matches the first letter of destination state capital.
The IN number represents how many other state capitals end with the first letter of the current capital. These are ways in to the state.
The OUT number represents how many other state capitals start with the last letter of the current state's capital. These are the ways out.

Whew. Now for the return trip:

Washington (Olympia) - 1 in: Sacramento. 5 out: Albany, Annapolis, Atlanta, Augusta, Austin

Maine (Augusta) - 5 in, 4 out (same as above)

Maryland (Annapolis) - 6 in, 6 out (same as above)

Oregon (Salem) - 4 in: Annapolis, Columbus, Des Moines, Indianapolis. 3 out: Madison, Montgomery, Montpelier

Vermont (Montpelier) - 1 in: Salem, 2 out: Raleigh, Richmond

Virginia (Richmond) - 3 in, 3 out (same as above)

Iowa (Des Moines) - 4 in, 6 out (same as above)

Oregon (Salem) - 4 in: Annapolis, Columbus, Des Moines, Indianapolis. 3 out: Madison, Montgomery, Montpelier
Uh oh, she's been here before ...

Wisconsin (Madison) - 1 in Salem. 1 out: Nashville

Tennessee (Nashville) - 7 in: Austin, Boston, Charleston, Jackson, Lincoln, Madison, Trenton. 0 out: :(

The pattern of travel involves the following:

She can only travel to states with where the trailing letter of the current state's capital matches the first letter of destination state capital.

As such, I believe your cousin has currently found herself in:

Tennessee (Nashville)

Here's the explanation of her trip to Washington:

writing it now...

The pattern of travel involves the following:

She can only travel to states with where the trailing letter of the current state's capital matches the first letter of destination state capital.

As such, I believe your cousin has currently found herself in:

Tennessee (Nashville)

Here's the explanation of her trip to Washington:

Georgia (Atlanta) - 5 in: Augusta, Columbia, Helena, Olympia, Topeka. 4 out: Albany, Annapolis, Augusta, Austin

Maine (Augusta) - 5 in: Atlanta, Columbia, Helena, Olympia, Topeka. 4 out: Albany, Annapolis, Atlanta, Austin

Maryland (Annapolis) - 6 in: Augusta, Atlanta, Columbia, Helena, Olympia, Topeka. 6 out: Sacramento, St. Paul, Salem, Salt Lake City, Santa Fe, Springfield
Sacramento is a shortcut

Illinois (Springfield) - 4 in: Annapolis, Columbus, Des Moines, Indianapolis. 3 out: Denver, Des Moines, Dover
Dover is another shortcut

Colorado (Denver) - 4 in: Springfield, Richmond, Concord, Hartford. 2 out: Raleigh, Richmond

Virginia (Richmond) - 3 in: Denver, Dover, Montpelier. 3 out: Denver, Dover, Des Moines

Delaware (Dover) - 4 in: Concord, Hartford, Richmond, Springfield, 2 out: Raleigh, Richmond
Here's the mistake. It should be North Carolina instead of South Carolina
North Carolina (Raleigh) - 3 in: Dover, Denver, Montpelier. 4 out: Harrisburg, Hartford, Helena, Honolulu

Connecticut (Hartford) - 1 in: Raleigh, 3 out: Denver, Des Moines, Dover

Iowa (Des Moines) - 4 in: Concord, Hartford, Richmond, Springfield. 6 out: Sacramento, St. Paul, Salem, Salt Lake City, Santa Fe, Springfield

California (Sacramento) - 4 in: Annapolis, Columbus, Des Moines, Indianapolis. 2 out: Oklahoma City, Olympia

Washington (Olympia) - 1 in: Sacramento. 5 out: Albany, Annapolis, Atlanta, Augusta, Austin