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Future gamma-ray experiments, such as the e-ASTROGAM and AMEGO telescopes, can detect the Hawking radiation of photons from primordial black holes (PBHs) if they make up a fraction or all of dark matter. PBHs can analogously also Hawking radiate new particles, which is especially interesting if these particles are mostly secluded from the Standard Model (SM) sector, since they might therefore be less accessible otherwise. A well-motivated example of this type is axion-like particles (ALPs) with a tiny coupling to photons. We assume that the ALPs produced by PBHs decay into photons well before reaching the earth, so these will augment the photons directly radiated by the PBHs. Remarkably, we find that the peaks in the energy distributions of ALPs produced from PBHs are different than the corresponding ones for Hawking radiated photons due to the spin-dependent greybody factor. Therefore, we demonstrate that this process will in fact distinctively modify the PBHs' gamma-ray spectrum relative to the SM prediction. We use monochromatic asteroid-mass PBHs as an example to show that e-ASTROGAM can observe the PBH-produced ALP gamma-ray signal (for masses up to ~60 MeV) and further distinguish it from Hawking radiation without ALPs. By measuring the gamma-ray signals, e-ASTROGAM can thereby probe yet unexplored parameters in the ALP mass and photon coupling.