The phrase "start her job" can refer to her entire period of employment (that is, her getting hired) or her shift for a particular day.
The phrase "used to" takes an infinitive, so in the first one, "used to" can't apply to "started". (If you take out "drink tea" and just say "She used to started her job", that would be grammatically incorrect). Since "started" is thus not a habitual action, it must refer to her getting hired. So your first example means "Before she got hired, she used to drink tea", and implies that she stopped drinking tea after she was hired.
In your third sentence, both readings are possible: "starting" could be a habitual action preceded by drinking tea, or it could be a one-time process of being hired. It can be read as "She used to (drink tea before starting her job)" or "She (used to drink tea) before starting her job"; that is, it could mean that both drinking tea, and then starting her job, were things she repeatedly did, or it could mean that drinking tea was something she repeatedly did, and all of those instances of drinking tea occurred before one instance of getting hired.
With your second sentence, only the "She used to drink tea before each shift" reading works. Since this is the least ambiguous wording, you sthis is the best one out of the three.
You could also say "She used to drink tea before starting her job each day." @ThePhoton 's suggestion of "she used to drink tea before going to work." or just "She used to drink tea before work".