We present accretion-disk structure measurements from UV-optical reverberation mapping observations of a sample of eight quasars at 0.24<z<0.85. Ultraviolet photometry comes from two cycles of Hubble Space Telescope monitoring, accompanied by multi-band optical monitoring by the Las Cumbres Observatory network and Liverpool Telescopes. The targets were selected from the Sloan Digital Sky Survey Reverberation Mapping (SDSS-RM) project sample with reliable black-hole mass measurements from Hbeta reverberation mapping results. We measure significant lags between the UV and various optical griz bands using JAVELIN and CREAM methods. We use the significant lag results from both methods to fit the accretion-disk structure using a Markov chain Monte Carlo approach. We study the accretion disk as a function of disk normalization, temperature scaling, and efficiency. We find direct evidence for diffuse nebular emission from Balmer and FeII lines over discrete wavelength ranges. We also find that our best-fit disk color profile is broadly consistent with the Shakura & Sunyaev disk model. We compare our UV-optical lags to the disk sizes inferred from optical-optical lags of the same quasars and find that our results are consistent with these quasars being drawn from a limited high-lag subset of the broader population. Our results are therefore broadly consistent with models that suggest longer disk lags in a subset of quasars, for example, due to a nonzero size of the ionizing corona and/or magnetic heating contributing to the disk response.