Who saved the second largest number of Jews in WW2?

1st place by a massive margin, far more than all the rest put together, was the Catholic Church under Pius XII.  Historians believe around 850,000 were saved by Catholic institutions.

2nd place will come as a surprise to most: it was General Franco's Spain.  Franco is of course a bogeyman on the secular left.  The left are sore losers and hold the cultural high ground at the moment: government, academia and the media.  Franco is a bogeyman merely because he beat them, so the following facts remain largely untold.  If it does come up, the secular left will do its best to play it down by attributing some implausible, sordid motive to Franco.  In reality, one needs to look no further than the fact that Franco's own ancestors were Marranos (Spanish Jews who converted to Christianity to avoid expulsion).

Franco's Spain saved more Jews from the concentration camps than any other country.  The consensus of historians is that Spain saved between 20,000 and 35,000 Jews by letting refugees pass through the country.  Plus another 5,000 Jews were saved by Spanish diplomats abroad.  Franco refused to hand over Spanish or foreign Jews to the Nazis and he allowed Jews and others with Portuguese visas to transit Spain.  Despite his aversion to Zionism and "Judeo"-Freemasonry, Franco clearly did not share the rabid anti-Semitic ideology of the Nazis.

Early in WW2, a contingent of Spanish doctors travelling in Poland were informed of the Nazi extermination plans by Governor-General Hans Frank, who was under the misimpression that they would share his views about the matter; when they came home, they passed the story to Admiral Luís Carrero Blanco, who informed Franco. Another piece of this jigsaw could have been Admiral Canaris, the head of German Military Intelligence. Canaris had been appalled by the Nazi's brutality he had witnessed in Poland and he had worked with Franco during the Spanish Civil War, and the two men had become friends.

In the first years of the war, the Jews were mainly from Western Europe, fleeing deportation to concentration camps from occupied France, but there were also Jews from Eastern Europe. Throughout World War II, Spanish diplomats of the Franco government extended their protection to Eastern European Jews, especially in Hungary.  Jews claiming Spanish ancestry were provided with Spanish documentation without being required to prove their case and either left for Spain or survived the war with the help of their new legal status in occupied countries.

Diplomats discussed the possibility of Spain as a route to a containment camp for Jewish refugees near Casablanca but it came to naught because of the lack of Free French and British support. Nonetheless, control of the Spanish border under France was relaxed, and thousands of Jews managed to cross into Spain (many by smugglers' routes).  Almost all of them survived the war.  The American Jewish Joint Distribution Committee operated openly in Barcelona.

Shortly afterwards, Spain began giving citizenship to Sephardic Jews in Greece, Hungary, Bulgaria, and Romania; many Ashkenazic Jews also managed to be included, as did some non-Jews. The Spanish head of mission in Budapest, Ángel Sanz Briz, saved thousands of Ashkenazim in Hungary by granting them Spanish citizenship, placing them in safe houses and teaching them minimal Spanish so they could pretend to be Sephardim, at least to someone who did not know Spanish.  Toward the war's end, Sanz Briz had to flee Budapest, leaving these Jews open to arrest and deportation.  An Italian diplomat, Giorgio Perlasca, who was himself living under Spanish protection, used forged documents to persuade the Hungarian authorities that he was the new Spanish Ambassador, and continued Spanish protection of Hungarian Jews until the Red Army arrived.

I believe, but cannot trace the source of this story, that there was an agreement with the Nazi for Spanish diplomats to issue just 200 papers.  These papers were numbered consecutively, and Briz and Perlasca got round this hurdle by issuing the numbers randomly, and simply never issuing the number 200 - and by this simple ruse were able to save thousands.

Franco's Benevolence to the Jews Continued Long After the End of WW2

In 1967, after Egypt had been pulverized in the Six Day War by Israel, Egypt turned in vengeance on its Jewish community, throwing at least 425 Jewish men into jail. Other Arab countries also began terrorizing their Jews.

The 75 year old General Franco sent instructions to his diplomats in Egypt, Lebanon, Libya, Tunisia and Morocco to help all Jews, whether Sephardic or otherwise, and to provide Egyptian Jews with documents to leave.  Following these instructions, the Spanish ambassador in Cairo, Angel Sagaz, helped over 615 families escape over the next two years, even transferring some of their jewellery (which Egypt forbade them to take with them) out of the country in Spanish diplomatic bags.