I don't know that Christianity is necessarily connected with the free market in the United States. Right-wing Christianity is. But it's important to point out that while they aren't as vocal, there are a lot of self-identified Christians on the US Left. The reason they aren't as prominent and the reason the right-wing Christians are more prominent is that on the right they're part of what many would call the Reagan Coalition.

As President Reagan put it:

Ronald Reagan used to describe the Republican Party as a kind of three-part coalition. Each part was primarily motivated by its pet issue: national defense, free-market economics, traditional values. Together these comprised a "three-legged stool" that supported the party. Saw off any of the legs and the stool collapsed--taking the party down with it.

At this time he was saying this, the issues could be boiled down to a few - prayer in schools, the right to life, and a general moral decline that they opposed. While I'm certain they had economic concerns, those weren't the top issue, and the general premise of a coalition is that you're aligning yourself with other factions you don't always agree with to achieve power. While that's not a statement you can prove, I feel justified using that definition when looking at other coalitions. FDR's New Deal Coalition united trade unions, ideologues of the political left, and rural area voters through his programs such as extending electricity to the rural areas and farm housing bills. You could also look at any number of Parliamentary "Coalition Governments" where there is no defined winner, so parties unite despite differing parties to gain power.

You asked: Was there ever a time that the major Christian religions in the US leaned more heavily towards social programs?

Probably, but if the overarching question is how right-wing Christians became associated with the free markets despite many of the bibles dictums saying "care for the poor" then I can really only answer with this - the Christians on the Left aren't a voting bloc. And if they're not in a coalition, their Christianity wasn't something that made it to the party platform plank.

Keep in mind, that a lot of what the Bible indicates people should do for the poor is already a general plank in the Democratic Partys platform and has been since FDR. The Right/Left Divide will generally argue "Should the government, or the people be charitable?" Since that's off-topic I'll leave it there, but it's important to point out that Christians don't need to highlight this aspect (caring for the poor) in their political desires because (and I'm assuming here) many of their wants are built-in.

Really, up until the Voting Rights Act in the Johnson Administration, the current day Bible-Belt in America was in the Democratic Party. And FDR himself when asked about his ideology said "I am a Christian and a Democrat. That is all."

You wrote: I believe part of it is rooted in the religious importance of liberty and agency (and the implication that big government limits those) but wasn't sure if there was more depth to it.

If I can ad-lib here for a second "Politics is a hell of a drug." It's unclear why liberty and agency would have been a real catalyst, except for where there was a political division (e.g. Prayer in Schools). Freedom of Religion is in the US founding documents, and as far as I can tell has never come close to being an issue for the Christian Religions (potentially Catholics, but that's a separate issue). And in FDR's Four Freedoms, Freedom to Worship is included. So it's not as if the Democratic Party is anti-religion, even if there are politicians over time who have tried to accuse one party or another of being godless.

I also keep citing FDR because in many ways he's the bedrock of the modern Democratic Party. But both Jimmy Carter and Barack Obama made faith a central part of their general rhetoric.

The right-wing Christians adherence to a free-market is almost certainly rooted in two thigs:

1. They had a coalition with the "fiscal" leg of Reagan's stool, and that leg was very free market
2. Reagan was very pro-capitalist and very anti-Soviet.

And so it can be said that I did my due diligence, a central theme of the Soviet Union was a lack of religion. Here you can likely find the best alchemy of why Christians fully adopted the free-market.

• They got power in the US by aligning with the fiscal group
• They had a common cause with the National Defense group in being anti-Soviet/Communist since they were anti-religious
• Those two legs supported their positions

There are two other areas I think are important that wouldn't have come up in either your question or the writing above.

1. You could make an argument that Reagan's three-legged stool had a progenitor in the Nixon Administration
2. The moral decline mentioned above has one foot in "Law and Order." Something Nixon campaigned on heavily.

These two issues are really just two sides of the same coin. At the 1968 Democratic National Convention, there was absolute bedlam. And it was televised. (I don't like Wikipedia links but this isn't really a historical event up for debate). It wasn't anything like today. The President of the United States, then LBJ, called Chicago Mayor Richard Daley and said what amounted to "Get those hippies off my lawn." While I'm sure that wasn't quite what he said, and I'm certain it was a lot more vulgar it ended with the two wings of the party (a coalition falling apart) fighting with each other. Radicals vs. authoritarians on television in one party.

This tied in somewhat to moral decline. The establishment were no better than the Communists with their iron-fisted rule, and the hippies believes in free sex, drugs, and the devils music. You might say, looking back all these years, Nixon had the political genius to see how to build a coalition, and Reagan sold it. That sounds a bit demeaning so I want to clarify that this is a major gloss-over, but the broad strokes I'm painting are a starting point for further reading.

Finally, the potential undoing of the Reagan Coalition

From FDR to Reagan was about 50 years. Which is about where we are now. Coalitions are generally built by the powerless, and fall apart when everyone at the top has become accustomed to ruling and getting their way. Generally speaking they think everyone sees it their way and they don't watch the rear flank.

While I can't predict the future, there are some noticeable flashpoints between the stool legs. And one of the issues I keep seeing in discussions is this: Was the Bargain between the Christians and the Libertarians a deal the Christians are losing? For example, here in an interview with Ross Douthat (a Christian journalist with the NYT), Ezra Klein asks (I'm paraphrasing because I listened awhile ago) "Did the Christian right have to agree to not crusading against say, porn, because of the coalition." It was likely more articulate than that, but just as the New Deal Coalition fell apart, it's not unreasonable to wonder aloud about the free market mentality and what it's provided for the Christian right.