  Blazars are some of the brightest UV and X-ray sources in the sky and are
valuable probes of the elusive warm-hot intergalactic medium (WHIM; $T{\simeq}
10^5-10^7$ K). However, many of the brightest blazars$-$called BL Lac objects
such as 1ES1553+113$-$have quasi-featureless spectra and poorly constrained
redshifts. Here, we significantly improve the precision of indirect redshift
constraints for blazars based on the edge of the $\rm{H\,I}$ Ly$\alpha$ forest
observed in their UV spectra. We develop a robust technique to constrain the
redshift of a $z<0.5$ AGN or blazar with a $1\sigma$ uncertainty of
${\approx}0.01$ using only the position of its highest-redshift Ly$\alpha$
absorber with $\log N_{\rm{H\,I}}/{\rm cm^{-2}} > 12.6$. We use a large sample
of 192 AGN/QSOs at $0.01\lesssim z\lesssim0.45$ that have high-quality COS FUV
spectra to characterize the intrinsic scatter in the gap between the AGN
redshift and the edge of their Ly$\alpha$ forest. We present new COS NUV data
for 1ES1553+113 and confirm its redshift of $z=0.433$ using our technique. We
apply our Ly$\alpha$-forest-based redshift estimation technique to nine
additional blazars with archival ${\it HST}$ UV spectra, most of which are key
targets for future X-ray missions. Our inferred redshift constraints improve
estimates for two BL Lacs (1ES1118+424 and S50716+714) and are consistent with
previous estimates for the rest. Our results emphasize the need to obtain
further UV spectra of bright blazars, of which many have uncertain redshifts,
in order to maximize the scientific value of future X-ray WHIM observations
that will improve our understanding of galaxy evolution.
