Would delaying the demographic transition work? I found a good chart on the net...
...and it looks like it might be doable without much additional territory if you can keep the USA in Stage 2 and 3 for much longer - you basically need to stop the US from industrializing (or at least slow down the general timetable of industrialization) so that large families are still required for things like small industry and farming, yet improved medical care allows the death rate to go into serious decline, which lets the population just keep on climbing. How you can delay American industrialization is the real problem in that case, as all the geographical ingredients are there. Possibly something along the lines of, say, a worse American Civil War that degenerates into bushfire/insurgency status later on, basically a simmering conflict? Something like that could cause attacks on railways and mines and make the wealthy reluctant to invest in the USA, which would slow down industrialization, which means the USA stays in the earlier half of the demographic transition for longer which will result in a higher population later on. Combine that with something like, say, a later invention of vulcanized rubber (slowing down the arrival of the cheap condom) and you have something that seems like it could be part of the solution, if not the solution in itself.
Hmmm...perhaps some kind of American Luddite movement forms before the Civil War, which gets the backing of the Southern planter class (as machines would be fighting against slavery's economic value) with renewed conflict with Britain causing people to associate machinery and the processes of industrialization with European powers and European tyranny, possibly with a religious aspect towards it as well, causing a popular resentment towards the processes of industrialization...which, looking back from the cosy modern world, weren't exactly that nice in the first place, what with the mangled children coming out of textile mills and the like. That would slow it down (it wouldn't stop it, because obviously someone is going to come along eventually, smoke a cigarette and ask "what the fuck are we doing" and shuts it down), which could result in a a fair number of extra people joining the United States. Probably borderline ASB, but a thought all the same.
I don't see you doing this without a PoD pre-1900, though. By that point the USA was already well into the demographic transition and the birthrate was starting to drop. You need that rate to stay as high as possible for as long as possible if you are to hit your goal.