There is no doubt that a worsening of economic and social conditions in Cuba will provoke an increase in legal and illegal immigration, mainly to the US. It can get out of hand, but not easily.
In previous situations, Boca de Camarioca in the 60’s, Mariel in the 80’s and the last one in the mid 90’s, always the Cuban authorities had, in some justified way, allowed it to go on until the US authorities were forced to some kind of agreement. But it was always first provoked by rigid US policies that did not take in consideration the consequences of such policies.
I think that the Cuban Government and Party have the means and political tools to avoid such a situation today, but it is obviously a possibility.
As you have been probably able to watch, the destruction in Pinar del Río, Holguín and Las Tunas, but not only in these provinces, has been enormous and it has hit private houses in the worst way. Just as much, it also hit agricultural production and electrical energy infrastructure.
There is already a certain scarcity of sweet potatoes (bonitato), malanga (I don’t know the name in English), bananas and plantain, fresh pork and goat meat and others basic food in the “agromercados” and it will get worse. There is also a problem in the CUC ("dollar") stores to get cooking oil (you can only get soya oil), tomatoe paste, canned fish and meat, frozen chicken, cheese and fresh meat.
The only solution is a rapid recuperation of agricultural production. The private agricultural sector is the main producer, but can not do it by itself, It is needed that the state farms under their various forms of organization increase production, something they have been unable to do in 50 years, without huge investment in machinery, fertilizers, insecticides and other inputs that are no longer available.
As the destruction is shown by TV, people get surprised to see how poor the houses were before the hurricanes, and slowly everybody is starting to realize that it will take decades to bring housing to, even, the previous poor situation.
An immediate political result of the destruction is the solidarity and unity of purpose that brings among the people. This is probably true in every country. In Cuba it is even more so, as the Government and Party had created a solid organization that includes more and more people to people solidarity, to confront hurricanes and heavy rains. Also because of the quick and effective response to start reconstruction, that includes, even, cultural groups with well known artists performing in the more severely affected areas. Of course the limited resources are the main problem.
But, is difficult to predict how the mood will change as the reality of an even poorer country, with even more economic and social problems, takes hold slowly of people's minds.
So, yes it is almost certain that there will be an increase of emigration. Whether it will be massive and illegal, depends of many factors. Of course the impact of a limited in time lift of the embargo, or the increase in remittances and traveling, will help to avoid that this problem gets out of hand. As has happened in the past, I doubt the US Administration will take in consideration the consequences of their fanatic anti-evolution policies. Even without massive emigration, it is always safer for you not to provoke problems to your neighbor that can affect you in the long run.