In recent years, programmable Rydberg atom arrays have been widely used to simulate new quantum phases and phase transitions, generating great interest among theorists and experimentalists. Based on the large-scale density matrix renormalization group method, the ground-state phase diagram of one-dimensional Rydberg chains is investigated with fidelity susceptibility as an efficient diagnostic method. We find that the competition between Rydberg blockade and external detuning produces unconventional phases and phase transitions. As the Rydberg blockade radius increases, the phase transition between disordered and period-3 density-wave ordered phases changes from the standard three-state Potts to the unconventional chiral universality class. As the radius increases further, a very narrow intermediate floating phase begins to appear. This result provides a concise physical picture to illustrate the numerical results. Compared with previous studies, this work provides more evidence for non-conformal quantum phase transitions in programmable quantum simulators from the perspective of quantum information, showing that fidelity susceptibility can be used to study commensurate-incommensurate phase transitions.