Weather in Israel early November… trekking south or north Israel

Hi, we found a bargain flight to Tel Aviv early November and like to know if the weather is still warm then usually? We want to escape the grey and rainy November here in Berlin for two weeks. Probably traveling to Akko, Jerusalem and Death Sea maybe Negev. We are both passionate hikers and not sure if we should do some trekking in the North or better in Negev… maybe in the South it's warmer and we also could do a site tripe to Petra? We are going to rent a car in Tel Aviv and could we leave the care for a bus-trip to Petra at the border? Any suggestions are welcome. Thanks.