The politics of a larger stimulus is difficult.
A lot of it is that frankly there just wasn't an appreciation yet for how bad the recession was getting. There's the oft attacked "Job Impact of the American Recovery and Reinvestment Plan" report from Obama's incoming economic council on the potential impact of the stimulus. It predicted between 3-4 million jobs being created by the end of 2010, with unemployment (with) the stimulus peaking just below 8 percent in 2009, and (without) peaking at 9 percent in 2010. Which was wrong. What occurred with the stimulus was unemployment peaking at 10 percent and not even getting blow 8 percent until the middle of 2012. We hadn't had a recession as deep or as long lasting as the Great Recession in a long time, and it showed it how the policy was crafted; anything over a trillion dollars was seen as being loony.
The other thing is that the bill relied a lot of some Republicans and more Conservative Democrats to back it. When it came up for a vote in February you had just 58 Democrats + Independents, which includes the likes of Joe Lieberman, Evan Bayh, and Max Baucus. Specter didn't switch to the Democrats until April, while Franken wasn't seated until July of that year due to the drama in Minnesota. What we got was, for the time, the largest we could have hoped for.
To change these I think you need an earlier recession, a bigger Democratic majority, or both. An earlier recession so by the time the election happens and the new administration is formulating its policy objectives there's a clearer understanding that this isn't going to be a normal downturn, and a bigger majority to allow Democrats to override concerns about the deficit or to focus on tax cuts.
As for impact. Hard to say? A lot depends on just how big it is. The original being in the $850B area, versus plausibly $1.2T, or $2T which is what we now think we probably needed. I don't think $1.2T changes much fundamentally. It helps, obviously, but you're talking changes along the margins. Perhaps unemployment doesn't hit 10 percent, peaks a bit sooner, and the recover takes hold quicker. Unemployment didn't really seem to start steadily going down until 2011; with a few extra hundred billion say that starts in 2010 and is more steady. I'd be cautious though since even with a $2 trillion stimulus I don't know how quickly or rapidly we could expect a 'full' recovery. We had experienced a severe financial crisis and skirted financial meltdown, there's not going to be an easy way to avoid the impacts of a deep recession, and the recovery will take time.
The other important question is what the money is being spent on and when. Some of the criticism of the original stimulus is that, for all the touting of shovel ready projects, there just wasn't much to spend on when job losses were at their most severe. This is highlighted in that, as far as I know, only about a fourth of the Stimulus' disbursements occurred in 2009. If you want to add another trillion to it, where to actually spend that sort of poses a challenge. More state aid, surely, but that's not worth a trillion even if you stretch it over a year or two more. You could do more with tax cuts or tax credits, but that offers a limited benefit to someone who has already lost their job. We could always spend more on infrastructure, but its not really suited for 'immediate' spending. Which is to say with a larger stimulus I think we need to get a lot more creative about how we want the money spent, and a lot more realistic about for how long we want the money spent.