In my opinion, 'Secular Buddhism' can be problematic since often adherents to this new group (such as Stephen Batchelor, who believes purity of mind is not possible) often misrepresent the Buddhist teachings due to no aspiration to practise Buddhism at a high level. They are like Protestant Christians that seek to modify the Buddhist teachings to suit themselves.
While there is nothing wrong with not believing in post-mortem rebirth, there is an understanding of karma & 'rebirth' that does not involve believing in an afterlife. To write-off 'karmic inheritance' is not particularly wise since there is such thing as karmic inheritance (such as when a murderer ends up in prison for their actions).
I would recommend books by monks such as Bhikkhu Buddhadasa, Ajahn Chah, Ajahn Sumedho, etc, because these demonstrate both a proper understanding of Buddhism & the 'here-&-now' approach you are interested in.
It is important to understand the Buddha himself taught his teachings are about the 'here-&-now', what in Pali is called: "Sanditthiko, Akaliko, Ehipassiko, Opanayiko and Paccattam Veditabbo Vinnuhi". Therefore, being 'secular' is not contrary to the true Buddhist teachings.
However, if you are interested in a 'watered-down' Buddhism that may help you with ordinary life then 'Secular Buddhism' might be suitable.
I personally would recommend the following books if you want to learn about true Buddhism while not interested in 'post-mortem' rebirth & other unverifiable belief systems:
more books by these authors are here: http://www.dhammatalks.net/
You are also welcome to read my posts or ask me questions on this chatsite since these are always 'secular'. There is a here-&-now interpretation for the majority of the vast teachings attributable to the Buddha, including those about 'rebirth'.