The southern part of the central can be very hot in September. We've spent parts of the last 3 years in Barquinha, right where the Tejo (Tagus) turns east to head into Spain. That region, and Santarem district just south of it but north of Lisbon, can feel very much a "heat trap". The first fall, 2017, I was there from mid-August to mid-September, and it was never less than 30 degrees and, for a week or so in the beginning, 40 at mid-day. I'm from the Pacific Northwest and that was pretty brutal for me. I wondered what I'd gotten myself into.
In 2018, if I remember correctly, it was 30-ish in early September before we arrived. Then we turned up. In mid-September, just after we arrived, we had another 40-ish heatwave that lasted 4 or 5 days, I think.
Last year, as others have noted above, it was 30-ish in the autumn. We had gone earlier in the year, and family matters meant that we had to cancel our planned visit in the fall.
At 30, walking on the unshaded dirt roads through fields, or walking along secondary roads, is not very pleasant. The coastal route should be much more pleasant. You'll get the ocean breezes. There is a not-very-high but significant range of hills between the coast and the Tejo "flatlands" where the central route south of Tomar runs, and curves around to block that heat just north of the river (i.e. at Vila Nova da Barquinha). Those hills trap the interior heat in the Tejo valley. It's not till you get over them, and past Tomar, that the heat moderates, if there happens to be a heatwave when you arrive.
Here, in Portuguese but easy to figure out, are the averages for Santarem, which is the regional centre with fairly representative weather for that whole area south of Tomar, which is what you want to consider. Unfortunately, their averages cover only to the year 2000. With climate change, it's probably safe to say that the new "averages" are close to the red line, which was the earlier days' extreme max.
www.ipma.pt
Bom caminho, whichever route you choose!