I really can help you on this because I have done a massive amount of research from the original Parish Records of Oxfordshire over the last two years. Firstly, Thomas Rodgers contains a 'd' (in spelling) in its original form. I am certain that your Thomas is my first cousin, four times removed. His father is another Thomas Rodgers as is his grandfather. This family originates from South Newington (in North Oxfordshire), near Banbury. We share a common ancestor, Henery Rodgers, who is my 5th Great Grandfather. He had his own sheep farm and died in South Newington in 1743. Alfred T. Willetts from Shenley, Buckinghamshire (who served in the American Civil War)is my third cousin, twice removed. His mother is Jane Ursula Rodgers (my second cousin three times removed) from Over Norton, Oxfordshire. The best bet is to find my public tree in Ancestry and then you can see what I have found out. Please contact me if you want to know any more detail.
Regards, Clive Rodgers