Washington (CNN) The Senate's razor-thin balance of power -- 52 Republicans and 48 Democrats -- means that just a few senators can tip the scale on whether a bill passes -- and some lawmakers are already embracing their potential influence.

As the GOP looks to take advantage of having Republican President-elect Donald Trump in the White House, party leaders have promised an ambitious first 100 days agenda. But to do so, they'll need to keep everyone in their party happy.

And senators on both sides are quick to realize they can gain leverage if a vote will come down to party lines. That provides opportunities for senators with pet issues to draw lines in the sand, for vulnerable senators up for re-election to get airtime, and for moderates on both sides to work across the aisle on compromise bills.

Some examples have already emerged.

Kentucky Republican Sen. Rand Paul voted against the motion to open up Obamacare repeal proceedings on Wednesday do to opposition to the underlying budget. Though Republicans still had the votes to proceed, Paul is lobbying his colleagues and working with members of the House to try to force his colleagues to oppose deficit spending, even if it jeopardizes Republicans' pledged Obamacare repeal.

Paul met Thursday morning with the House Freedom Caucus, a group of roughly 40 Republicans in the House who have caused headaches far bigger than their small numbers for Republican leadership. Paul left the meeting without firm commitments, though the group plans to continue to meet on it.

While on his way to make a similar pitch to his Senate colleagues, Paul said in response to a question that the Freedom Caucus could serve as a model for a small percentage of the GOP caucus having a big impact in the Senate, too.

"We're going to try. We'll see," Paul said.

Another Obamacare issue is the fate of efforts to defund Planned Parenthood, which is opposed by most Democrats. Two pro-abortion rights senators, Susan Collins of Maine and Lisa Murkowski of Alaska, won't commit to approving the bill with the Planned Parenthood provision in it, endangering GOP hopes of easily getting the Affordable Care Act repeal legislation through the Senate.

Other senators are also flexing their muscles on key issues. As Trump continues to deny and downplay Russia's role in hacking Democratic political groups in an effort to influence the election, as determined by the US intelligence community and third-party experts, Sens. John McCain and Lindsey Graham have been vocal in their condemnation of Russia's interference.

McCain and Graham are working with Democrats, including Maryland Sen. Ben Cardin, on legislation that would impose sanctions on Russia beyond those announced by the White House last month, which would force Trump to accept the sanctions or publicly revoke them.

McCain and Graham both emphasized that their work on Russia wasn't out of the mainstream. "I think there's a supermajority for what I'm talking about," Graham said when asked about how much influence a few senators could have.

But McCain acknowledged that the narrow Senate majority gives heft to a small voting bloc.

"That makes it more impactful," he acknowledged, while also emphasizing the bipartisan nature of the effort.

Tillerson nomination

The narrow majority has also allowed Republicans to hold sway over Trump's nominees.

When Trump nominated Exxon CEO Rex Tillerson as secretary of state, McCain and Graham joined Sen. Marco Rubio in expressing concerns and questions about Tillerson's friendly relationship with Russian President Vladimir Putin, with whom Tillerson has interacted for his business interests in Russia.

McCain has said since meeting with Tillerson some of his concerns have been answered and Rubio said Thursday he will meet with Tillerson early next week, but the nominee is sure to face tough questioning on the issue during his confirmation hearings on Wednesday.

The Senate Foreign Relations Committee, which will hear Tillerson's nomination, is one of the committees that only have a one-vote majority for Republicans this Congress -- meaning any unsatisfied Republican could vote against advancing a nominee to the floor and make the nominee's path more difficult. Membership on that committee includes Rubio; Arizona Sen. Jeff Flake, who is up for re-election this year, has sparred with Trump in the past and is likely to be targeted by Democrats; and Paul, whose foreign policy views tend to be more isolationist than his party's and who is against nation-building.

Rubio demurred when asked if he and his colleagues could seek to increase their influence by forming a bloc, as did Texas Republican Sen. Ted Cruz. The two close friends, along with fellow good friend Utah Sen. Mike Lee and blue-state Republican Nevada Sen. Dean Heller, have already teamed up on legislative advocacy in the first three days of this Congress.

"I'm not sure anyone around here's thinking in those terms," Rubio said. "By and large, we worked real hard to have a majority so we can undo a lot of the damage that's been done to this country over the last few years. So I've never heard or think there's any effort ongoing to create some sort of bloc to steer policy and create some sort of leverage."

Still, if three members voted in concert, they could prove a make-or-break bloc on legislation. And Cruz and Lee have not shied away from taking on members of their own party in the past.

Democrats as powerbrokers

And it's not just Republicans who could take advantage of the narrow majority.

On Thursday, 13 Democrats put out a statement appealing to Republican leadership on Obamacare, billing themselves as "moderate Democrats" open to working across the aisle, including Democrats such as North Dakota Sen. Heidi Heitkamp and Missouri Sen. Claire McCaskill, who are up for re-election in red states in 2018.

West Virginia Sen. Joe Manchin, a Democrat in a red state also up for re-election next term, has also been looking to reach across the aisle, skipping a meeting with President Barack Obama and Democrats on the Hill this week in favor of talking to both sides.

Manchin said he hopes Republicans also understand the importance of working with a few members of the other side, citing how many Democrats it will take them to overcome the 60-vote filibuster threshold.

"They understand, it's going to take eight. If they can hold all the Republicans at 52, then it will take eight of us to get anything moving," Manchin said. "So you're going to have to be working with 10, 12, truly sitting down with 10 or 12 moderate Democrats as you think that you want to move things forward."