We propose an ultrasensitive displacement measurement scheme to overcome the standard quantum limit (SQL) in the unresolved sideband cavity optomechanical system with nonlinear optomechanical coupling and squeezed light injection. By introducing the optimized quantum correlation, which is enabled by suitable choices of the squeezing angle, squeezing level, power of the probe light, and measurement angle of homodyne detection, the off-resonant displacement sensitivity reaches 6 dB below the SQL in linear optomechanical coupling. In contrast, displacement sensitivities with a coherent probe plus variational readout and squeezed probe plus fixed measurement angle (phase quadrature) are 2.6 dB and 4.6 dB below the SQL, respectively. By combining linear and quadratic optomechanical coupling, we show that the displacement sensitivity can be further improved to 9.6 dB below the SQL. Our results have potential applications in gravitational-wave detectors, quantum metrology, and the search for dark matter.