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Blazars are some of the brightest UV and X-ray sources in the sky and arevaluable 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 objectssuch as 1ES1553+113$-$have quasi-featureless spectra and poorly constrainedredshifts. Here, we significantly improve the precision of indirect redshiftconstraints for blazars based on the edge of the $\rm{H\,I}$ Ly$\alpha$ forestobserved in their UV spectra. We develop a robust technique to constrain theredshift 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 sampleof 192 AGN/QSOs at $0.01\lesssim z\lesssim0.45$ that have high-quality COS FUVspectra to characterize the intrinsic scatter in the gap between the AGNredshift and the edge of their Ly$\alpha$ forest. We present new COS NUV datafor 1ES1553+113 and confirm its redshift of $z=0.433$ using our technique. Weapply our Ly$\alpha$-forest-based redshift estimation technique to nineadditional blazars with archival ${\it HST}$ UV spectra, most of which are keytargets for future X-ray missions. Our inferred redshift constraints improveestimates for two BL Lacs (1ES1118+424 and S50716+714) and are consistent withprevious estimates for the rest. Our results emphasize the need to obtainfurther UV spectra of bright blazars, of which many have uncertain redshifts,in order to maximize the scientific value of future X-ray WHIM observationsthat will improve our understanding of galaxy evolution.

Improving blazar redshift constraints with the edge of the Ly$\alpha$ forest: 1ES 1553+113 and implications for observations of the WHIM