This is the 569th most frequent Russian word.
другого
"Of another" or "different."
'другого' functions as an adjective here, modifying the noun 'выбора' (choice) to mean a different or alternative choice.
У меня нет другого выбора.
I have no other choice.
'другим' (instrumental case of 'другой') is used here as an adjective to describe 'путём' (path), indicating the selection of an alternative or distinct course of action.
Она решила пойти другим путём.
She decided to take a different path.
'другого' here is used as a pronoun referring to a different object (another taxi) than the one previously implied or mentioned.