When to squeeze Pompeii into our itinerary

We stayed in Sorento, so I'm not sure about transportation to Amalfi Town (or Positano or the others) except the bus we took from Sorento to the Amalfi coast towns. My suggestion would be to do Pompeii on the way to Amalfi. You can take the high speed train (which can be booked in advance for a savings) to Naples, then we took the circumvesiviana from there to Pompeii Scavi. There is luggage storage at the train station. I've heard there are other transportation options between Naples and Pompeii, but we found the circumvesiviana just fine.

I think i would stop on the way to Rome

The journey from Florence to Amalfi is a long haul already and I would want to be at Pompeii early in day
It will still be hot in September and there is no shade

Assuming you are taking fast train from Florence to Salerno then ferry to Amalfi?

Reverse that and stop at Pompeii on way to Rome
You will have to walk a bit further to scavi entrance from that train station but you avoid the Circumvesuviana
A nice train ride after exploring Pompeii all the way to Rome will feel like heaven

The option to stop on the way just means more crowded travel onward to Amalfi.

I don’t think the fast Trenitalia train to Rome stops at the three modern Pompeii city stations. So you would have to change in Naples to the Frecciarossa service. Check me on that.

Number 3 is not advisable because it would take over two hours to get to Pompeii from Amalfi by public transportation. A car service with waiting time would be hundreds of euros. My guess is that a taxi to Pompeii would be €125. But you could hail a different one to go home and not pay for waiting time. Note that some residences in Amalfi could be a long walk to the bus or taxi.

It doesn’t matter whether you visit on the way to Amalfi or on the way to Rome. The key concept is avoiding the hottest and sunniest part of the day for your Pompei visit. I haven’t used the free luggage checking at Pompeii but legions of posters on this news board have nothing but good things to say about it. I’d recommend having a quick lunch at the cafeteria inside Pompeii because your travel day is not a good time for two hours of gourmet slow food. And you can’t re-enter Pompeii after exiting.

Edit: I don't have a budget idea for your luggage transport to Amalfi, but also think about how it reflects on luggage getting to Pompeii. Perhaps late September will be a little better, but if you choose the lowest cost option, it's almost impossible to squeeze a full-size suitcase onto the Circumvesuviana. Once you arrive in Sorrento (or for that matter, Salerno), you have to wait for a half-hourly SITA bus (after buying tickets at the nearby newsstand) and hope that the line for the bus is fewer people than will fit on the next bus. (It often is longer than that.) This won't be so bad in Salerno, if you go that way. But the luggage at Pompeii may be harder, if there is no luggage checking at the Pompeii-city Trenitalia station, you'll have to lug it to the Amfiteatro entrance of Pompeii, where there is another free bag check, and then back to the train station.

A substantial fraction of Amalfi or Positano guests hire a car service, or pay the fixed rate for a Naples taxi. We took the Circumvesuviana coming into Sorrento, and it was quite an ordeal. That, combined with the 15 Euro taxi half-a-mile to our Sorrento hotel, easily sold us on having our hotel concierge order a black car (air conditioned mercedes) to take us back to the Naples train station. 90 Euros, but that was ten years ago. Ferries might be possible, but you have to get to the ferry dock.

Note that you cannot wake up in Amalfi and fly home from Rome the same day. I think you can see why.