A hand wash will not remove the swirl marks, but it should be much better than using the automatic car washes. You will need a buffer to remove swirl marks. Meguiars makes a good one, and so does Porter Cable. Search online at Google for them. You can just get a cheap one at Target, Wal-Mart and use that. It won't be as good as the two I just named; but it can help. A good product for swirl removal is made my 3M. It's available at Pep Boys and online. 3M Swirl Mark Remover for Light or Dark Cars. Another product by 3M that's available at Wal-Mart, Autozone, Discount Auto Parts that can be used by hand (pain, but can help) is called 3M Finish Restorer. Good product. Also, Scratch X made by Meguiars applied by Buffer can help, as well as Meguiars #9 Swirl Mark Remover, Meguiars DAPC (Dual Action Polish Cleaner), or an AWESOME product by a company called S100 or P21S (same products with different names and prices). S100 Shine Enhancing Cleanser - Awesome by buffer or hand. A good glaze like Meguiars # 7 can be used after, and then finish it all off with S100 Paste Wax, which is awesome and the easiest wax to use. It shines like no other caranuba out there. Only downside with S100/P21s products is that they are for motorcycles (of course they can be used for cars) and can only found at Harley dealerships, or online at Harley's website, or at a good detail store. Turtle Wax makes a Swirl Mark Remover that might be cheaper than everything and maybe will help you. A good regimen would probably be;
Wash it, claybar (if the car feels real rough - Clay Magic brand), Swirl Mark Remover (anything I named above), Polish (3M Imperial Hand Glaze, Meguiars # 7 Show Car Glaze, other stuff), Wax (Meguiars #26 Liquid Wax, S100 Paste Wax, Klasse, Blitz, etc.) Hope I helped some.