  Precision measurements of rare particle physics phenomena (flavor
oscillations and decays, electric dipole moments, etc.) are often sensitive to
the effects of new physics encoded in higher-dimensional operators with Wilson
coefficients given by ${\rm C}/(\Lambda_{\rm NP})^n$, where C is dimensionless,
$n\geq 1$, and $\Lambda_{\rm NP}$ is an energy scale. Many extensions of the
Standard Model predict that $\Lambda_{\rm NP} $ should be at the electroweak
scale or above, and the search for new short-distance physics is often stated
as the primary goal of experiments at the precision frontier. In rather general
terms, we investigate the alternative possibility: ${\rm C} \ll 1$, and
$\Lambda_{\rm NP} \ll m_W$, to identify classes of precision measurements
sensitive to light new physics (hidden sectors) that do not require an
ultraviolet completion with additional states at or above the electroweak
scale. We find that hadronic electric dipole moments, lepton number and flavor
violation, non-universality, as well as lepton $g-2$ can be induced at
interesting levels by hidden sectors with light degrees of freedom. In
contrast, many hadronic flavor- and baryon number-violating observables, and
precision probes of charged currents, typically require new physics with
$\Lambda_{\rm NP} \gtrsim m_W$. Among the leptonic observables, we find that a
non-zero electron electric dipole moment near the current level of sensitivity
would point to the existence of new physics at or above the electroweak scale.
