As suggested, pumps help, by building up a guaranteed 7/7ths magma on the outward side of the pump, as long as there's a supply on the inward, at least. My tapping of magma channels is generally by smoothing then fortifying the wall between the magma source and the magmaduct I've just dug[1], which gives plenty of time for the stoneworker to wander away, if I've connected diagonally into it. But it's slow, when I do that. As you don't (unless you time it
very well) have two or more separate fortifiers working side-by-side, the gap is only one-wide, however.
But, it's possible (though too late for your situation, as it is for the installation of pumps unless you have somewhere else with a shut-off valve) that you could make several individual breaches into the magma-pipe, separated and floodgated so that each of them can be made without danger from the others pouring through, all leading into the 'main' magmaduct. I'm not entirely sure if this would feed as I think it would, but with 7/7ths present behind a number floodgates all opening onto the wider magmaduct.
As a not-perfect example:
#######oooooooooo
########ooooooooo
####++xHooooooooo
####X###ooooooooo
+++++##oooooooooo
+++++XHoooooooooo
+++++##oooooooooo
####X###ooooooooo
####++xHooooooooo
########ooooooooo
#########oooooooo
#= Rock, + = dug out rock, X and x = Floodgates, H = Fortifications, o = Magma
Dig and place the floodgates before doing anything else (if you're not using any other tricks to prevent stuck dwarves, dig only up to the "X" squares first, fill them with floodgates, then open them so you can dig and emplace the 'x' floodgates).
Once you have the two 'x' gates and the central 'X' one in place, and operational, one by one open them, set to smoothing and fortifying (my preference) or just dig through (more dangerous, but might give better flow, I'm not sure what the current state is of Magmaflow dynamics, I've played safe for so long). Slam them shut immediately you get magma-flow (and, obviously, the dwarf is out of the way), and let any magma that got through evaporate as necessary, then work on the next one in the same way.
Once that's all set, and all is ready, close the non-central 'X' gates and open the 'x' gates to fill the small feeder channels up to to the non-central 'X's. Now you have three floodgates ready to unleash magma into a 3-tile-wide magmaduct.
If it works as I anticipate (ICVWBW) you'll get a progress of magma roughly the same speed of a 1-wide gap supplying a 1-wide duct, only tripled in volume.
It certainly will stop the biggest problem with supplying a 3-tile-wide duct (aqua or magma) with a 1-tile-wide entrance hall, the fact that unless there is pressure behind the liquid concerned, the 'front' of the flow (and perhaps some tiles behind) will be 1/7th and liable to evaporate spontaneously even while you're willing the liquid to reach the other end. It won't stop it[2], but might well slow the flow down, in a way that the 3-wide supply won't be so subjected to.
That point effectively ninjaed by ikkonoishi.
[1] And smoothed, and cleared of useful debris/wall material, and replaced the mined-out walls constructed replacements made of blocks of the same stuff as the 'base' material surrounding it.... I spend too much time on that, though.
[2] Although I've previously 'fanned out' a liquid to spill over a large enough plain to get an equilibrium point of spread beyond which it has so much 1/7ths that evaporate that it doesn't tend to travel any further, when deliberately trying to 'lose' liquid without atom-smashing or sending through edge-fortifications or into 'sink' aquifers/etc)