Please suggest me new ipad smart cover compatible back cover
I preffer tpu cases
Thanks

Apple has quietly added a new accessory for the second-generation and new iPads in the Apple Store today: the Smart Case. This is a separate product from the original Smart Cover that debuted with the second iPad. This was not announced at the WWDC Keynote.
The new Smart Case is basically the Smart Cover, except with a matching back shell. The product is made of the same polyurethane material and is available in six colors,including light gray, dark gray, blue, green, pink, and red.
Because of the included back cover, the Smart Case comes at a $10 premium over the polyurethane Smart Cover. It can also be engraved for free if ordered online.
Cost is $49.00
I know it's too late for your needs, I'm just putting it out there.

Information found on App Advice.

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would you suggest a bit more cheaper cover? It cost too much for me.

I suggest you find it on ebay. There are many many of them, cheap but beautiful, around $15 usually.

I found this one to work great, the price is very reasonable too.

Amazon.com: Luvvitt DOLCE Slim-fit Smart Cover Companion / Compatible TPU Case (BACK COVER ONLY) for iPad 2 and iPad 3 - Transparent Black: Computers & Accessories

i would definetly go with a Devicewear Union back cover all the way. I recently did a full review on one and have used it personally myself every since. Its the perfect back cover hands down in my opinion. Check out my signature for a link to the review if your interested.

Cimo smart cover back! I got mine through Amazon and it's amazing!!!!

ArmorSuit from Amazon's back cover should be compatible with the smart cover. Hope this helps. Good luck

check this out - Happy Owl Studio -
Happy Owl Studio iPad 2 "The Leather Shell" Protector Case review
It might say for ipad2, but also works for ipad3.

Check here on, Best iPad 3 cases and covers | Apple - CNET News. Best cases collection for iPad 3.