  For a standard $\Lambda$CDM universe with a power-law primordial power
spectrum, the discrepancy between early- and late-universe measurements of the
Hubble constant continued to grow and recently reached $5.3\sigma$. During
inflation, local features in the inflationary potential often lead to
band-limited features in the primordial power spectrum, hence breaking the
power-law assumption in the derivation of the Hubble tension. We investigate
whether such inflationary "glitches" can ease the Hubble tension. The recently
released \emph{Planck} temperature and polarization data and the 2019
SH0ES+H0LiCOW joint constraint on the Hubble constant are combined to drive a
blind Daubechies wavelet signal search in the primordial power spectrum, up to
a resolution $\Delta \ln k\sim 0.1$. We find no significant detection of any
features beyond power law. With 64 more degrees of freedom injected in the
primordial power spectrum, the Hubble tension persists at a $4.9\sigma$ level.
