First do a survey of what are the contents of the packed archive. Most likely the vast majority are binary libraries which perl depends on. Logically, in this case, your best option is to install a complete Perl in your target system, if you can. Additional benefit is that these libraries can be reused instead of each packed archive using their own.

You can install Perl and modules in your target system but still have your program packed and run by PAR which will not include any Perl dependencies. This will result in a much smaller archive, hence faster extraction. See Perl-interpreter-only,-without-core-modules: of pp and Perl-with-PAR.pm-and-its-dependencies-installed: of pp. In this way you have the best of both worlds, (1) Perl installed in your target system once, and (2) you can distribute your program(s) as single executables of very small size as many times as you like.

If you want to extract once and use forever, then you can use saner filenames with -T or --tempcache (see -T,-tempcache of pp) to specify something else instead of an ugly hash. Most importantly, you can change the first part of the extract-to directory (i.e. the /tmp/...). See PAR::Environment and PAR_GLOBAL_TEMP etc.

If you are in Linux, you can extract into a temporary RAM-based dir: as root, mount -t tmpfs -o size=500m tmpfs /myramdir and then set PAR_GLOBAL_TEMP to /myramdir. Your program will live there until reboot or unmount. That will be faster though ephemeral.

Also, there is the option of running perl through a docker linux container. Theoretically talking, your container should include perl and all module dependencies. Your program can be inside or outside the container.