A semi-coherent search is conducted for continuous gravitational waves from stellar oscillations in Hercules X-1 with data from Observing Run 3 of the Laser Interferometer Gravitational-wave Observatory, using an established hidden Markov model algorithm to track stochastic wandering of the signal frequency. The primary goal is to implement and test several algorithmic enhancements relevant to future searches, including a second-order projected semi-major axis correction, an orbital eccentricity correction, an adjustment to the width of a search subband, and propagation of the electromagnetically measured orbital parameters with a nonzero orbital period derivative. The enhancements are validated by injecting synthetic signals into Gaussian noise and improve the sensitivity depth by $\approx 14\%$ for a Hercules X-1-like target. In the 60–$200\, {\rm Hz}$ band, the pilot search yields four candidates that survive a five-level veto procedure, whose terminating frequencies are $76.41\, {\rm Hz}$, $109.47\, {\rm Hz}$, $144.40\, {\rm Hz}$, and $189.24\, {\rm Hz}$. The frequentist upper limit on the maximum detectable wave strain at 95% confidence is arbitrarily set at 174 Hz and found to be $9.3\times {10}^{-26}$, assuming a uniform prior on polarization.