No.

First, you have to look at the background of Rhee and the events leading up to the April Revolution.

Rhee was definitely a pro-Korean guy. He spent his whole life working towards Korean independence from Japan, spending most of the teens in the US where he studied and got his PhD, the 20's in Shanghai as the President of the government in exile, and then from the mid 20's, 30's and 40's in Hawaii. He was impeached in the mid 20's from his position for abuse of power and there were different factions vying for different goals (communists and anti-communists, anti-Japanese and pro-Chinese, all working together when their interests overlapped and fighting against each other when they didn't.)

When the Japanese surrendered, the government in exile was headed by Kim Gu (President) and Kim Kyu-sik (Prime Minister). These two gentlemen had to make their own way back to Seoul and didn't arrive for a few months. Syngman Rhee flew in on MacArthur's personal plane. Kim Gu was interested in a unified Korea and he met with Kim Il-sung prior to the Korean War to negotiate on reunification. Rhee was an anti-communist through and through who would have started the Korean War on his own terms to reunify Korea if the US had supplied him with the weapons necessary, which is why they were so woefully unprepared for the northern invasion.

All of this is to say that Rhee was not the most popular of leaders in 1945. He had the backing of the US because he had spent so much time in the US hobnobbing with DC politicians and they knew who he was. Kim Gu and Kim Kyu-sik were unknowns and they didn't toe the anti-Communist line that Rhee did. I should note that Kim Kyu-sik was favored by John Hodge, the commander of USAMGK as one of the leaders, but Rhee pushed hard for a UN decision which Kim Gu and Kim Kyu-sik opposed since the North wouldn't participate. After Rhee took power, Kim Gu was assassinated in 1949 Ahn Do-hee, who claimed in 1992 that he was ordered to do so by the head of Rhee's security service, Kim Chang-ryong. Ahn was then assassinated in 1996 by a follower of Kim Gu. It's important to remember that Kim Gu had murdered people before, notably Josuke Tsuchida in 1896, whom he suspected of helped to assassinate the Korean Queen Min. During the Japanese colonial era, lots of people resorted to murder and assassination, so we shouldn't dwell too much on the morality/legality of it.

Once Rhee was in power, he ordered the suppression of the Jeju Uprising. The Korean police/military with the assistance of the US military murdered men, women, and children on Jeju Island, as well as in Yeosu/Suncheon (the uprising there was in response to the government response in Jeju). Then, during the Korean War, the Rhee government summarily executed anyone remotely suspected of being a communist. Whole villages were killed as the southern forces retreated. Also, while evacuating Seoul, the government blew the Seoul bridges over the Han River as people were crossing it to escape the northern advance, killing scores of civilians.

We can easily say, by 1953, Rhee was a murderous strongman.

But none of this really pushed the April Revolution.

With the Korean War over, both North and South Korea worked on rebuilding. In 1956, the US gave (in constant 2009 dollars) a little over 2 billion dollars in cash and about 5.5 billion in military assistance. These numbers trended downward from a high in 1956, but the US consistently gave aid to Korea. After the war, Korea had modest GDP growth, however, from 1958 to 1960, the GDP per capita decreased. The government was ineffective and the economy was stalling. A large contributor to this was a financial stabilization program forced on the economy in 1957 and 58. Due to large influxes of foreign money, prices rose drastically and the government moved to stabilize prices, resulting in the slowdown. the "Miracle on the Han" that we know about didn't begin until the 60's under Park Chung-hee and the military government.

Politically, Rhee was very unpopular in 1952 and he should not have won re-election. At the time, the President was elected by the National Assembly, not directly by the people. Rhee pushed to change the law in favor of a direct election, which was rejected. So he used his power and had his rivals arrested, and then pushed the changes through the Assembly and won re-election with over 70% of the vote. In 1956, he had the Constitution changed to allow the incumbent (him) to run for re-election an unlimited amount of times. In 1960, Rhee had one opponent arrested and executed, and his other opponent died of a heart attack before the election, so Rhee won with 90% of the vote.

Enter the April Revolution!

The election for vice President was a separate affair and Rhee wanted his hand picked successor, Lee Poong-ki, to win. Unfortunately, this guy was running against Chang Myon, former ambassador to the US and a much more popular guy. Oddly though, Lee won the election by a margin of 4:1.

All of these issues came to a head with this election. The people had been under Rhee's oppressive rule since 1948. His government had murdered citizens in Jeju and Yeosu, then they murdered suspected leftists and communists during the war. After the war, the corruption enriched a few and economic growth for the masses was, in the good years, slow, and in the bad years, negative. The government clamped down on dissent and simply calling a neighbor a communist was enough to get them in serious trouble.

After the election, students came out to protest. In Masan, a port city just west of Busan and Gimhae, protests turned violent as the police fired on the protesters who responded by throwing rocks. A few weeks later, the body of a high school student, Kim Ju-yul, was found in the Masan harbor. The government reported that he drowned but people didn't believe it and they forced their way into the morgue. His head had been split open by a tear gas canister, an injury most likely caused by the police firing the canisters directly at the people. It has since come out through legislative investigation and a trial that the police were under orders to kill protesters and that the chief of the area had Kim's body weighted with rocks and tossed in the harbor so he wouldn't be found.

Rhee's administration attempted to blame communist agitators and censor the news, but it had already been picked up by the AP and the people knew what had really happened. This led to more protests culminating in the final protests of April 19th where the government fired on student protesters in Seoul. Rhee resigned a few days later, Lee Ki-poong's son murdered him and himself in a murder-suicide, and the government collapsed.

With all of this in mind, could Rhee have come back to power?

No. He didn't have the support. The First Republic collapsed and the Second Republic was created only two months later by the opposition party who won the elections. The role of president, so badly abused by Rhee, was reduced to a ceremonial role and Chang Myon, the opponent to Lee Poong-ki in the VP election, became the Prime Minister and the head of government.

Why would the press keep their mouth shut about Rhee?

Korean press, even today, has always been a dangerous job. Publishing reports critical of the government or people in it could easily get a writer or publisher killed, either "legally" with an accusation of communism/northern sympathy, a quick trial, and an execution, or illegally with someone simply murdering them as they walked home (See Kim Gu, Lyuh Woon-hyung). The fear that Rhee would come back was non-existent, but his political party/cronies could have remained in government and, even as a minority party, made life difficult for anyone who wanted to criticize Rhee. It's important to remember that South Korea has the National Security Law which is very vague. The Hankyoreh, a left wing paper established in 1988, had it's managing editor and a director arrested in 1989(!) for sending reporters to cover North Korea. The three big dictators (Chun, Park, Rhee) used the NSL very liberally to squash dissent, and it's still the law to this day, so the Korean press has (and still) always been very careful about what it says or does until they are sure they won't get shot for it.