The best solution would be using cookies since it will not affect your cache/full page cache (if you have it set). So, all you have to do is to:
1. Make an observer on checkout_cart_add_product_complete event to grab that custom attribute value and save it in a cookie with:
$cookie = Mage::getSingleton('core/cookie');
$cookie->set('cookiename', 'cookievalue' ,time()+86400,'/');
2. Create javascript file which you'll attach to <catalog_product_view> handle, and which should read that cookie value and put it into an appropriate input field.
I'm not sure that there is extension available which solves this issue for you, but this is technical approach which can be used by any skilled developer to implement such solution.
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