\subsection*{{\sc \large Supplementary Material}}
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\vspace{-3mm}
\section*{A.~Size of the PBH}
\vspace{-2mm}

The Schwarzschild radius of a black hole is given by
\begin{equation}\begin{aligned}
r_{\rm BH}=\frac{2GM_{\rm BH}}{c^2}\simeq 4.5{\rm cm}\left(\frac{M_{\rm BH}}{5M_\oplus}\right)~.
\end{aligned}\end{equation}
In Figure \ref{fig1} we provide an exact scale image of a $5M_\oplus$ PBH. The associated DM halo however extends to the stripping radius $r_{t,\odot}\sim8$AU, this would imply a DM halo which extends roughly the distance from Earth to Saturn (both in real life and relative to the image).


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\begin{figure}[h!]
\vspace{1mm}
\includegraphics[height=9.35cm]{Fig9cmBH}
	\caption{Exact scale (1:1) illustration of a $5M_\oplus$ PBH. Note that a  $10M_\oplus$ PBH is roughly the size of a ten pin bowling~ball.}\label{fig1}
\end{figure}

\newpage
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\section*{B.~Dark Matter Freeze-in via a $Z'$}

To demonstrate a viable freeze-in model with potentially detectable indirect detection signals we consider a specific implementation in which the Standard Model is supplemented by two new ``hidden sector'' states, the DM particle $\chi$ with mass  $m$ and a mediator $Z'$ with mass $M_{Z'}$. We assume a sizeable interaction $gZ'\overline{\chi}\chi$ for $g\sim\mathcal{O}(10^{-2})$, with mass ordering $M_{Z'}>2m$, and that the hidden sector only couples to the Standard Model fermions $f$ (with mass $m_f$) via $\lambda Z'\overline{f}f$, which involves a tiny coupling  $\lambda\sim10^{-12}$. 
 
Here we relabel the mediator $Z'$ (from $\phi$ in the main text) and highlight that it can naturally arise from a  kinetic mixing portal \cite{Holdom:1985ag} involving a new U(1)$'$ gauge symmetry via the Lagrangian term $\Delta{\mathcal L}=\delta B_{\mu\nu}X^{\mu\nu}$
where $X$ is the $Z'$ field strength and $B$ is the hypercharge field strength, where $\delta$  parameterises the mixing induced by states  (with masses $M$ and $M'$)  charged under both U(1)$'$ and hypercharge $\delta\propto \frac{g_Yg_X}{32\pi^2}{\rm log}\left(M^2/M'{}^2\right)$. In motivated models this can lead to couplings appropriate for freeze-in \cite{Dienes:1996zr,Gherghetta:2019coi} and one can identify the freeze-in coupling $\lambda\simeq\delta g_{Y}$ where $g_Y$ are Standard Model fermion hypercharge couplings. For further details on $Z'$ freeze-in portals see e.g.~\cite{Chu:2013jja,Chu:2011be,Elahi:2014fsa,Heeba:2019jho,Foldenauer:2018zrz}.

The initial  $Z'$ and $\chi$ abundances are negligible but populations are generated via the portal interaction $\lambda Z'\overline{f}f$.
 The $Z'$ abundance evolves according to \cite{Hall:2009bx}
\begin{equation}\begin{aligned}
\dot n_{Z'}+3n_{Z'} H\simeq  \int & {\rm d}\Pi_{Z'}  {\rm d}\Pi_{\overline{f} }{\rm d}\Pi_{f}  (2\pi)^4\\
 &\delta^4(p_{Z'}-p_f-p_{\overline{f}})|\mathcal{M}|^2_{Z'\rightarrow \overline{f}f}f_{Z'}^{\rm eq}
\end{aligned}\end{equation}
where $f_{Z'}^{\rm eq}={\rm exp}(-E_\chi/T)$ is the equilibrium distribution. 

The generated abundance of $Z'$ is  \cite{Hall:2009bx}
\begin{equation}\begin{aligned}
Y_{Z'}=\frac{135}{8\pi^3g_*^{S}\sqrt{g_*^{\rho}}}\left(\frac{M_{\rm Pl}\Gamma_{Z'}}{M_{Z'}^2}\right)
\end{aligned}\end{equation}
where $\Gamma_{Z'}$ is the partial width involving bath states. 

Since we take $M_{Z'}>2m$ and $g\sim\mathcal{O}(10^{-2})$ any $Z'$ produced decays promptly to DM pairs and thus the resulting DM relic density is $Y_\chi\approx2Y_{Z'}$ for $\lambda\ll g$. This can be related to the observed quantity
\begin{equation}\begin{aligned}
\Omega_{\rm DM}&\simeq
 0.2 \left(\frac{m}{100~{\rm GeV}}\right)
\left(\frac{\lambda}{6\times10^{-12}}\right)^2
\left(\frac{10~{\rm TeV}}{M_{Z'}}\right)~,
\end{aligned}\end{equation} 
where we have parameterised $\Gamma_{Z'}\simeq \lambda^2M_{Z'}/8\pi$. 

The feeble coupling $\lambda\ll1$ is characteristic of freeze-in models and is actually required to satisfy the consistency condition $Y_{\rm DM}(T)\ll Y_{\rm DM}^{\rm eq}$(T). If $\lambda$ is too large the hidden sector enters equilibrium ($Y_{\rm DM}\sim Y_{\rm DM}^{\rm eq}$) and then subsequently freeze-out dynamics sets the DM relic density.
However, since $\lambda$ is very small, direct and indirect detection is typically challenging. For the $Z'$ model the s-wave annihilation cross-section for $m_f\ll m<M_{Z'}$ is \cite{Alves:2015pea}
\begin{equation}\begin{aligned}
\sigma_0\simeq\sum_f \frac{N_c \lambda^2}{2\pi }
 \left(\frac{2g_{\chi v}^2   m^2}{(M_{Z'}^2-4m^2)^2 } + g_{\chi a}^2 \frac{m_f^2}{M_{Z'}^4}\right),
\end{aligned}\end{equation}
where the sum runs over Standard Model fermions and $g_{\chi a}$ and $g_{\chi v}$ are the axial and vector couplings to $\chi$. Then eq.~(\ref{xc}) is obtained from the above in the limit $ m\ll M_{Z'}$. In the main text we highlight that if $f_{\rm PBH}$ is relatively large, indirect detection may be a viable route to search for this class of models in certain parameter ranges.

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\vspace{3mm}
