We note that $ 2013 $ has the property: since the sum of the digits of 201, $ 2+0+1=3 $, is divisible by $ 3 $, $ 201 $ is itself divisible by $ 3 $. We must now check whether any odd-numbered years before $ 2013 $ have the property. $ 200 $ is not divisible by the prime $ 7 $, so $ 2007 $ does not have this property. Likewise, the sum of the digits of $ 200 $ is $ 2 $, which is not divisible by $ 3 $, so $ 200 $ and $ 9=3^2 $ are relatively prime. Clearly $ 2011 $ does not have the desired property since every natural number is relatively prime to $ 1 $. Thus, the first odd-numbered year after $ 2006 $ that has the desired property is $ \boxed{2013} $.