By hand you can polish, it will just take forever. Try Meguire's Scratch X after washing and clay barring. Work the product in well. Machine polishing is simply more effective due to the speed at which you can do it and the more aggressive pad selections you can choose from. After removing some of the minor swirls, your best bet is to fill-in the remaining swirls via a glaze. A traditional glaze is a non-abrasive liquid that fills-in minor swirls and scratches within a paint surface. It can be seen as an alternative to polishing, but it still is not as effective as polishing because you will not be able to remove some water spots and the deeper swirls in the paint. Try this stuff:
http://www.superiorcarcare.net/clvamowaxhag.html
Wax your car after applying the glaze. Oh, and, I used to use 3M Imperial glaze...it works decently well but it does not create as glossy of a finish as the Clearkote products, and it is a pain in the a-- to remove.