We present a new optical transmission spectrum of the hot Jupiter HAT-P-1b based on two transits observed with the Double Spectrograph (DBSP) on the Palomar 200-inch (P200) telescope. The DBSP transmission spectrum, covering a wavelength range from 3250 to 10007 Å, is consistent with that observed with the Hubble Space Telescope (HST), but the former has a finer spectral resolution. The DBSP spectrum alone reveals the presence of a pressure-broadened line wing for Na, the line cores for both Na and K, and tentative evidence for H2O. We obtain consistent results from the spectral retrieval analyses performed on the DBSP-only data set and the DBSP, HST, and Spitzer combined data set. Our retrievals suggest a mostly clear atmosphere for HAT-P-1b, with a cloud coverage of ${22}_{-3}^{+5}$ % that is dominated by enhanced haze. We derive subsolar abundances for Na, K, and C, and subsolar to solar for O. Future observations with James Webb Space Telescope and ground-based high-resolution spectrographs should be able to not only confirm the presence of these species but also stringently constrain the formation and migration pathways for HAT-P-1b.