In Spanish, the situation where you keep talking after the meal has ended with the people you shared your lunch or dinner is called "sobremesa" (literally over-table or after-table). Is there a word in English to mention that situation?
Being fluent in both languages, I would have to come to the conclusion that, short of @Hot Licks suggestion of 'table talk', there really isn't a perfect English substitute. As mentioned, 'table talk' has no sense of order, whereas the sobre in 'sobremesa' conveys a sense of 'over-and-above', 'in addition to', or, in terms of time, 'after'. Also, it may just be due to the fact that, culturally, the portion after the meal in which people often talk and joke is much less a vital part of the meal itself in English-speaking countries than in their Spanish-speaking counterparts.
There is a word for witty, philosophical chat at the dinner table, but it's taken directly from the Greek: deipnosophy. Someone who is particularly adept at such conversation is a deipnosophist. But you won't find one English speaker in ten thousand who knows these words.
A bit less obscure, but still rare, one could refer to a "post-prandial" chat or conversation. "Post-prandial" is actually used by doctors in reference to testing one's blood glucose level after a meal, but more typically they recommend testing "pre-prandial" (before lunch or dinnerl) or "fasting" (before breakfast) levels.