  Local $H_0$ determinations currently fall in a window between $H_0 \sim 70$
km/s/Mpc (TRGB) and $H_0 \sim 76$ km/s/Mpc (Tully-Fisher). In contrast, BAO
data calibrated in an early $\Lambda$CDM universe are largely consistent with
Planck-$\Lambda$CDM, $H_0 \sim 67.5$ km/s/Mpc. Employing a generic two
parameter family of evolving equations of state (EoS) for dark energy (DE)
$w_{\textrm{DE}}(z)$ and mock BAO data, we demonstrate that if i)
$w_{\textrm{DE}}(z=0) < -1$ and ii) integrated DE density less than
$\Lambda$CDM, then $H_0$ increases. EoS that violate these conditions at best
lead to modest $H_0$ increases within $1 \sigma$. Tellingly, Quintessence and
K-essence satisfy neither condition, whereas coupled Quintessence can only
satisfy ii). Beyond these seminal DE Effective Field Theories (EFTs), we turn
to explicit examples. Working model agnostically in an expansion in powers of
redshift $z$, we show that Brans-Dicke/$f(R)$ and Kinetic Gravity Braiding
models within the Horndeski class can lead to marginal and modest increases in
$H_0$, respectively. We confirm that as far as increasing $H_0$ is concerned,
no DE EFT model can outperform the phenomenological two parameter family of the
DE models. Evidently, the late universe may no longer be large enough to
accommodate $H_0$, BAO and DE described by EFT.
