Rydberg atomic (RA) receivers represent a revolutionary quantum technology for wireless communications, offering unprecedented sensitivity beyond conventional radio frequency (RF) antennas. However, these receivers detect only signal amplitude, losing critical phase information. While reference signals generated by a local oscillator (LO) can assist in phase recovery, existing modulation schemes designed for conventional systems perform poorly with this quantum detection mechanism. This paper introduces a breakthrough LO-aware adaptive modulation (LOAM) scheme specifically developed for RA receivers that dynamically adapts to complex fading channel coefficients. LOAM maximizes the minimum amplitude difference between constellation points, ensuring optimal detection performance. The innovation employs an adaptive co-linear constellation architecture aligned with the combined phase of reference signal and channel coefficient. For strong reference signals, LOAM generates symmetric constellation points centered at origin; for weak signals, it adopts non-symmetric distributions. The paper mathematically derives the threshold governing these operational regimes. Simulation results reveal the transformative impact of LOAM, demonstrating performance gains exceeding 45 dB over conventional modulation schemes, including quadrature amplitude modulation (QAM), phase-shift keying (PSK), and pulse-amplitude modulation (PAM).