Klopp fundamentally likes to build sleeping giants back into giants. He doesn't like prima donnas or those who don't take things serious enough at the right times. He is the antithesis of Real Madrid or Barcelona.
If, and it's a big if, he goes back into club management, it will be to build up another sleeping giant. If AC Milan were as crap as they were a few years ago, that club would probably be his thinking, but as they have been winning more recently he probably wouldn't pick them. Plus his philosophy of football would have possibly less impact on Italian football than other places as they love to defend from the back and are solid in that Italian philosophy of football.
He will more than likely go to the German International scene and try to build the sleeping giants of International football as they are woeful at the moment.
He will be mainly at home, it won't be every day at the training ground. He can go watch some matches of up and coming footballers in Europe. Spend time with family and discovering new hobbies that he probably never knew he liked, kinda thing. Plus still be involved with football.
I really can't see him doing anything other than the Germany job.
Unless he went back to Mainz or Dortmund, but Klopp generally doesn't go back, he goes forward.