In 2014, I arrived in Rome from the US, and then flew to Palermo on a separate ticket. Because this connection was not "protected" and there was no way to know how long immigration and baggage claim would take, what I did was buy two tickets. One left for Palermo three hours after my Rome arrival, and one left ten hours after. This way, I planned to take one plane and miss one plane, and the combined price of the two advance purchase tickets was lower than the cost of a last minute ticket on my travel day (a Monday - prices were really jumping for unrestricted tickets on that day).
In the end, I made my three hour "connection" with time to spare - but of course, I could just as easily have missed it.
When I did this, there were several airlines flying the Rome to Palermo route, all with multiple flights a day. At the time, Alitalia's "international" website https://www.alitalia.com/en_en had the same prices as their Italian website for these domestic flights, but it was in English while the Italian website was only in Italian. The US website had much higher prices - for the exact same flights.
Of course, you will need to check airlines and schedules carefully for your travel dates, as even if you're able to take your trip, I expect service to be much less than before. Use Skyscanner for this: https://www.skyscanner.com/
As for the procedure (again, before coronavirus), Alitalia flights from the US to Rome arrived at Terminal 1, while all other carriers used Terminal 3. Departures to Palermo were from Terminal 1 for Alitalia, and Terminal 2 for budget carriers like Ryanair. It's a 10 minute walk from T3 to T1 (we did it outdoors because it was a nice day, but I believe there's an indoor way if it's raining). Past security in T1 there was a Telecom Italia Mobile (TIM) store, so we could buy SIM cards; that, plus breakfast, is how we occupied the time we had before our flight, since we had no hold up at immigration or baggage claim or T1 security. We were able to print our boarding passes for our Rome to Palermo flights before we left the US, due to the time difference; these days, I'd look to have the airline app on my phone, so I could have my boarding pass on that (saves time not having to get a boarding pass on arrival in Rome).