I work at a non-major US carrier, which many pilots feel is definitely inferior to the Big 6: (Delta, UAL, AA, FedEx, UPS, SWA). When we called fatigued, our schedulers tell us that we will have 12 hours of crew rest minimum, and to call when we get to the hotel and tell them how many hours of rest we want. Period.
We normally fly 1 leg per day, sometimes two. We have a very few circumstances of 3 leg days. We had a relatively long 3 leg trip in Asia, that occurred frequently. We had a number of fatigue calls, so that trip is now done by two crews: The first crew flies two legs, and the second crew the third leg. The schedule was changed because of the fatigue calls.
I hope working conditions will improve for you.
God Bless, and Namaste...