Fall

You'll start in the north due to cooler weather. As you go further south, day temperatures drop slowly. When you get to the Negev in early October or later, average day temperatures are not higher than 26 degrees celsius.

An alternative is to start from Tel Aviv and hike north. Come back to Tel Aviv by bus and head south.

Winter

I'd start in Arad and go south. Come back from Eilat and go north from Arad. More rain is expected, and cooler temps. Cold weather is relative: 15-20 c in the south 12-17 in the north. 

Spring

Your best option is to start the hike in late February to mid March from Eilat arrive to Tel Aviv and continue the hike from the north. From Tel Aviv take a bus to Kiryat Shemona (842). From Kiryat Shemona take bus number 55 or 58 to Dan or to Nimrod (13:30 and 16:40) and hike south to Tel Aviv. Average temperatures in March in the Negev are 14-23°C with very little rain, making March an ideal time to start. When you reach the area of Tel Aviv in late March or early April, the spring blossom is still at its peak and rain is becoming less likely.

Numerous local hikers choose the north as their starting point. If you start in the north, rainy days are more frequent in March.

I don't recommend to start in April from the north. You will end hiking in late May in the desert and it is very hot. 

Another alternative is to start in March in the area of Tel Aviv and hike south to Eilat. From Eilat take a bus to Tel Aviv and a bus to Kiryat Shemona in the north and hike south.

Summer

Avoid the desert. I'd say don't hit the trail in summer at all.

But if you do, then start from the north and make sure that you rest between 09:30-15:30 in the shade. Take more water with you: 5-6 liters / day for drinking.

One section is very hot in the North: Nahal Amud. Do not hike in one day from the Meron junction to Migdal - Kinneret (Sea of Galilee)