By the elections in June the TISZA Párt become the second largest party, with about 30%. Picrels are the freshest data.

Pic #1 made by a company close to the govt/Fidesz. It shows the EP election results in the first column at each party, and the current polling in the second.
- Fidesz is as popular as four months ago
- Tisza Párt's popularity has grown - remember this is a semi-conservative, pander to everyone, get the opposition together party
Between the two there's 10% difference.
- DK-MSZP-P is the remnants of the socialist party just above the 5% eligibility line (for parliamentary seats)
- Mi Hazánk, right radical party, separated from the Jobbik, from those who did not want to team up with the left-liberal opposition, again right at the threshold
- Kétfarkú Kutya Párt (KKP) - the joke party, under the treshold, perhaps they could get in perhaps not.
- Momentum - liberals, down the drain
- the rest down the drain.
According to them strictly taken those numbers 4 parties could get into the parliament.

Pic #2
From a pollster close to oppostion. They also show the preference in two columns (rows actually), but the light blue is the whole population (all voters), the dark shows the sure voters.
- Fidesz - 28% all voters, 39% sure voters
- Tisza - 26% all, 37% sure

One could think the Hungarian Revolution of '56 had great impact to the other Eastern Block countries. It is hard to measure but it seems the opposite is the truth. Officialy it was labelled as a counter-revolution and the rest of the opinions were suppressed. There were little to no samizdat activity that could give a voice, censorship ruled.
We already had some understanding with Poland, but Romania was openly hostile, Czechosolovakia ditto (with serious anti-Hungarian chauvinist overtones in both countries). Officially in the latter country as a direct reaction there was a "secret" memorandum that declared support for the leadership of Antonin Novotny, securing his position of First Secretary of the Communist Party of Czechoslovakia which he held since 1953, then he became president in 1957, held both offices till the Prage Spring removed him.
As our Revolution gained some of the ideas from the de-Stalinization and some wind too, Novotny's reign started with that - but as a counterweight. He seems he was more of a hardliner, who put effort into halting any thawing processes, and his "socialist" constitution of 1960 was even more formally communist than the previous one they accepted in 1948. So this "destalinization" went slow with occasional pushes from more reformist elements (such as writers, or the next First Secretary, Alexander Dubcek), eroding Novotny's popularity.
I can't say how wide was the popular support, but judging by the events that followed the later Soviet/Warsaw Pact invasion it seems that sympathizers were from all over the country from all level of the society - but not in any impactful numbers. People wanted changes but did not want to risk a more serious push for them. What we should see, that the changes were the product of the rise of an internal political opposition, and were introduced upside down, no armed uprising against the communist leadership happened unlike at Hungary.
In 1968 January Dubcek and a pal cornered Novotny, and Brezhnev (probably already probing the opinions of the leaders of the satellite states) decided to green light Novotny's retirement. The issue seemed to be decided but then Dubcek and many reformists started to talk nonsense about living standards, multi-party systems, democracies, building relations with the West and such. This compelled Brezhnev to engage them and tell 'em they should slow down a bit. They agreed, but The Head and Brow of the Soviet Union was not convinced, he saw that the process will lead to losing Czechoslovakia, and ordered the tanks of the SU and allied Warsaw Pact (Bulgaria, Poland, Hungary) countries' tanks rolling. No clash happened between the two sides, almost only civilians resisted, mostly with non-violent methods.
Dubcek had to go, he was replaced by Gustav Husak. All enacted laws and reforms were reverted, except the one that made Czechoslovakia a federation of two countries. Later some of the reforms were activated again as amendments if I understood correctly.

As for Hungary.
It appears that at the same time the Czechoslovak reforms started, the Kádár regime here introduced new economical practices, trying to make the country's economy actually work. The changes put forward in the neighbour were mostly political in nature, so our leadership started to pray to Saint Lenin that they keep them very "moderate". They feared if things were getting out of hands then our economical project will be scraped, or perhaps even the thoughts of resistance will be reignited here or at least Moscow will think that's happening too.
Since their fears seemed to become true, they decided to show they are loyal subjects of Moscow, and they can march along with the Warsaw Pact, right into Czechoslovakia, when finally Brezhnev decided at the military solution. Kádár even declared that if it comes to counter-revolution in Czechoslovakia, he will vote wholeheartedly for military intervention. Well he did the same with his own country so I don't see how could he not applied the same principle elsewhere.
From his opinion we could also deduce, that the Dubcek-turn, was went far enough for him to consider it a counter-revolution. However this is not the case. He gave the opinion to Brezhnev (on the day of the invasion, on August 21), that the events in Czechoslovakia are closer to the Polish events of 1956 than to the Hungarian counter-revolution of the same year. Still, Hungarian forces were participated the same.

Supposedly later in 1988-89 when our socialist system started to collapse they considered the moral Czechoslovakian lesson and example.