Upgrade linux kernel 3 to 4 in Debian 8

Add something like

deb http://mirror.one.com/debian/ jessie-backports main contrib non-free

to your sources.list. To install the 4.6 kernel, run:

apt-get update 
apt-get install -t jessie-backports linux-image linux-image-amd64

It might depend on a few other things that can also be found in backports, you might have to add those packages names to the command line explicitly. Apt will automatically track the versions in backports for the packages you install from backports, and not install anything from there unless you explicitly ask for them.

And after reading the entire question: It should be possible to leave the old kernel installed, and then grub should be configured to offer you a choice.