Sobremesa, the natural continuation of a good meal in good company has a long tradition in Spain, it does not come from the times of Franco's dictatorship. Having a dive in the Corpus Diacrónico del Español (my most sincere thanks to Guillermo Rojo for managing this titanic project) shows that "sobremesa" is very well documented prior to Franco's time. Let's work our way backwards from prior to Franco.

In 1936, we find the following text in Felipe Sassone's book "De sobremesa": Alcohol, allied with coffee, will unleash your tongue during the sobremesa, pleasure for philosophers, artists, and mundanes, bond of fraternal solidarity, cornerstone of your intelligence and of your ability to be sociable.

So, by 1936 the sobremesa was such a well established concept that books were written on the matter as shown by Felipe Sassone's example. It also indicates that it was not something exclusive to Spain, as Sassone was Peruvian. Indeed one may find plenty of references to sobremesa in Latin-American literature. Let's take a ship back to Spain while we work our way back in time. In 1898 the great writer Ángel Ganivet uses the word sobremesa as well:

El Eco a magazine he wrote in the blink of an eye and to make some money, for he had found Martina's purse in shambles. Pablo ad don Florentino came in the afternoon and stayed for lunch, and during the sobremesa it was agreed that the wedding be on Sunday morning, and that on the evening would the two travellers depart towards Barcelona, accompanied by the honest salesman from San Sebastián

We can keep making our way backwards and get to the famous "costumbrista" author Ramón de Mesonero Romanos, a great chronicler of Madrid's life during the 19th century through his diaries and his "Escenas y tipos matritenses", where we find this passage in 1850:

Later in the sobremesa, it was mandatory for the family to cross arms and surround the guest, in order to hear from his mouth the narration of the strange adventures of his pilgrimages, during which the papa couldn't help but to grow soft, the son but to grow enthusiastic, and the lady, if there was one, to leer at the outsider that better leave it at that

We still haven't advanced that much, we are still in the 19th century. Does that mean that sobremesa is a recent phenomenon? Nothing further from reality. Let's take a big jump and go to the Spanish Golden Age, and give the honourable Juan de Arce de Otálora, president of the Hall of Civil Matters of the Chancillería of Valladolid. In 1550, in his Coloquios de Palatino y Pinciano, we find this little gems:

This needs much brains; better leave it for sobremesa and shall we then dispute it with the bachelor. Let's see what he has to say. [...]

We have everything backwards, because we never speak nor have good conversation but in the sobremesa and after being full, when no relative is poor.

Twenty years prior, the great writer fray Antonio de Guevara, bishop of Mondoñedo, and a member of Charles V's council, for he was a wise man, illustrates a bit further on the matter of sobremesa, and how it can flow:

But now I ask what patience is necessary to endure it, or what heart to dissimulate it, that two or three or four people after eating for sobremesa, and taking a book in their hands, one says it is prolix; another says it speaks outside of logic; another that it is dark, another one that its language is bad; other one that everything it says is fictious; other says it is of no profit; another that it is curious; other one that it is malicious, so in the best case, the doctrine becomes suspicious and the author does not go unsullied.

All of this is, of course, just about the use of the word "sobremesa" to define to define the period after lunch, or even after dinner, where people gather round the table (or "over the table", which is what sobremesa means) and keep chatting for however long it takes, as it is just a matter of spending quality time with people we like. This custom is extremely ancient in and of itself, and goes right into the mists of time. The Romans had the convivium, which is another variant of the sobremesa, when people would dance, sing, or just chat after a meal. The same can be said of the greeks and their symposia, which is fundamentally the same.