  We derive a point-mass (nonspinning) frequency-domain TaylorF2 phasing
approximant at quasi-5.5 post-Newtonian (PN) accuracy for the gravitational
wave from coalescing compact binaries. The new approximant is obtained by
Taylor-expanding the effective-one-body (EOB) resummed energy and and angular
momentum flux along circular orbits with all the known test-particle
information up to 5.5PN. The -- yet uncalculated -- terms at 4PN order and
beyond entering both the energy flux and the energy are taken into account as
free parameters and then set to zero. We compare the quasi-5.5PN and 3.5PN
approximants against full EOB waveforms using gauge-invariant phasing
diagnostics $Q_\omega=\hat\omega^2/\dot{\hat\omega}$, where $\hat\omega$ is the
dimensionless gravitational-wave frequency. The quasi-5.5PN phasing is found to
be systematically closer to the EOB one than the 3.5PN one. Notably, the
quasi-5.5PN (3.5PN) approximant accumulates a EOB$-$PN dephasing of
$\Delta\Psi^{\rm EOBPN}\sim10^{-3}$rad ($0.13$rad) up to frequency $\hat\omega
\simeq 0.06$, 6 orbits to merger, ($\hat\omega \simeq 0.086$, 2 orbits to
merger) for a fiducial binary neutron star system. We explore the performance
of the quasi-5.5PN approximant on the measurement of the tidal polarizability
parameter $\tilde\Lambda$ using injections of EOB waveforms hybridized with
numerical relativity merger waveforms. We prove that the quasi-5.5PN point-mass
approximant augmented with 6PN-accurate tidal terms allows one to reduce (and
in many cases even eliminate) the biases in the measurement of $\tilde\Lambda$
that are instead found when the standard 3.5PN point-mass baseline is used.
Methodologically, we demonstrate that the combined use of $Q_\omega$ analysis
and of the Bayesian parameter estimation offers a new tool to investigate the
impact of systematics on gravitational-wave inference.
