  We present results of the magnetic susceptibility on high-quality single
crystals of EuPd$_2$(Si$_{1-x}$Ge$_x$)$_2$ for Ge concentrations 0 $\leq x
\leq$ 0.105 performed under varying hydrostatic (He-gas) pressure 0 $\leq p
\leq$ 0.5 GPa. The work extends on recent studies at ambient pressure
demonstrating the drastic change in the magnetic response from
valence-change-crossover behavior for $x$ = 0 and 0.058, to long-range
antiferromagnetic (afm) order below $T_{\text{N}}$ = 47 K for $x$ = 0.105. The
valence-change-crossover temperature $T'_{\text{V}}$ shows an extraordinarily
strong pressure dependence of d$T'_{\text{V}}$/d$p$ = +(80 $\pm$ 10) K/GPa. In
contrast, a very small pressure dependence of d$T_{\text{N}}$/d$p \leq$ +(1
$\pm$ 0.5) K/GPa is found for the afm order upon pressurizing the $x$ = 0.105
crystal from $p$ = 0 to 0.05 GPa. Remarkably, by further increasing the
pressure to 0.1 GPa, a drastic change in the ground state from afm order to
valence-change-crossover behavior is observed. Estimates of the electronic
entropy, derived from analyzing susceptibility data at varying pressures,
indicate that the boundary between afm order and valence-change crossover
represents a first-order phase transition. Our results suggest a particular
type of second-order critical endpoint of the first-order transition for $x$ =
0.105 at $p_{\text{cr}} \approx$ 0.06 GPa and $T_{\text{cr}} \approx$ 45 K
where intriguing strong-coupling effects between fluctuating charge-, spin- and
lattice degrees of freedom can be expected.
