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In this paper, we derive the early-time asymptotics for fixed-frequencysolutions $\phi_\ell$ to the wave equation $\Box_g \phi_\ell=0$ on a fixedSchwarzschild background ($M>0$) arising from the no incoming radiationcondition on $\mathcal I^-$ and polynomially decaying data, $r\phi_\ell\simt^{-1}$ as $t\to-\infty$, on either a timelike boundary of constant area radius(I) or an ingoing null hypersurface (II). In case (I), we show that theasymptotic expansion of $\partial_v(r\phi_\ell)$ along outgoing nullhypersurfaces near spacelike infinity $i^0$ contains logarithmic terms at order$r^{-3-\ell}\log r$. In contrast, in case (II), we obtain that the asymptoticexpansion of $\partial_v(r\phi_\ell)$ near spacelike infinity $i^0$ containslogarithmic terms already at order $r^{-3}\log r$ (unless $\ell=1$).  These results suggest an alternative approach to the study of late-timeasymptotics near future timelike infinity $i^+$ that does not assumeconformally smooth or compactly supported Cauchy data: In case (I), our resultsindicate logarithmic modifications to Price's law for each $\ell$-mode. On theother hand, the data of case (II) lead to much stronger deviations from Price'slaw. In particular, we conjecture that compactly supported scattering data on$\mathcal H^-$ and $\mathcal I^-$ lead to solutions that exhibit the samelate-time asymptotics on $\mathcal I^+$ for each $\ell$: $r\phi_\ell|_{\mathcalI^+}\sim u^{-2}$ as $u\to\infty$.

The Case Against Smooth Null Infinity III: Early-Time Asymptotics for Higher $\ell$-Modes of Linear Waves on a Schwarzschild Background