If we assume they're loaded in Mammoths, the largest dropshiop that can land, and that each dropship can carry 6,000 people, when you account for life support, that means that you would need 10,000 mammoths to carry just sixty million--and 100 times that number of people were moved. Now granted, even if we assume that it took five years and had the best organization (unlikely given the time) that is still an unimaginable feat--
I think the proposed time scale is a bit small; 10 to 20 years might be reasonable for the following reasons.
The Hegemony's population probably broadly started realizing it backed the wrong horse c2774 when fighting had been in earnest for a few years. Like the late Succession Wars, the Star League Civil War involved a small number of troops relative to the population, and the losses (1 billion civilians) were also proportionally small on the scale of the Hegemony and, say, 20th Century Wars. Depending on how Amaris and Kerensky mucked with civil transport in the Hegemony, there was an opportunity for anti-Amaris Terrans to start "visiting their mother-in-laws in the colonies" before the end of the Civil War. So that's several years of possible, low-rate emigration.
Second, there was a good 10 years between Terra's liberation (and it got liberated good and hard) and the start of the Succession Wars. In this period, life on Terra started among the worst of the Hegemony planets because of its severe liberation. Further, it would be painfully clear to Terrans - the core, head, and heart of the Hegemony - that the Hegemony's recovery was mostly politely termed a dumpster fire (as discussed in Jihad Hotspots: Terra). That's ten good years to sour the morale of Terrans beyond just "hey, I'm living in rubble." You could expect emigration from Terra in earnest in this period, and the attitudes lead to point 3.
Third, the start of the Succession Wars was not a simple, "Rarg, let's plunder the Hegemony!" The Houses didn't launch invasions of the Hegemony in 2787 in a vacuum, so to speak. There were years of build-up where - among other causes for the Hegemony invasion - the Houses found some Hegemony planets begging to join them. The first shots of the First Succession didn't sweep the Hegemony from side to side at once; it started with conflicts on the House borders and scraps over disputed Hegemony planets. In short, this was not a situation where, in 2787, Terrans suddenly halted their emigration. Rather, it likely took some time before it was clear that the rest of the universe was a worse place than Terra. So, you've got at least part way into 2788 (when Operation Silver Shield kicked off) for Terrans to continue their emigration. After Silver Shield ended, ComStar was eager to show what a fair and enlightened group it was (e.g., offering free tickets home for troops that had opposed ComStar on Terra), so it's possible that emigration continued. As noted in the ComStar SB, it was actually into the 29th Century before ComStar focused on Terran reconstruction specifically to placate Terrans, so they didn't necessarily fall instantly in love with ComStar. Emigration was reasonable until the First Succession War got truly bloody (2790s).
Therefore, 10 to 20 years of emigration at varying levels is reasonable, IMO.
Using 6,000-person Mammoths completing 12 round trips to another planet per year, a given Mammoth can move 72,000 people per year. In 10 years, it can move 720,000 people. You'd need 8,333 Mammoths to move 6 billion in 10 years. Assuming the average JumpShip involved had 3 collars, you'd 2,778 JumpShips for the full 10 years. That would be about 1/7th to 1/8th of the estimated 3025-era JumpShip fleets. Big, but on the other hand in the 2300s at least one individual owned 1000 JumpShips.
But that's the conventional approach. Using the 1-month round trips suggested above, the Mammoths only need to carry passengers for 2 weeks at a time. The consumable tonnage required by infantry bays is manageable for 2 weeks. For a generous 10 tons of infantry bay and consumables, you can carry 28 Terran refugees; a Mammoth wouldn't be strained to carry 84,000 people. (That's by game rules. DropShips are roomy for their tonnage, but I'm trying to imagine 84,000 people in a 200-meter metal balloon.)
84,000 refugees per Mammoth cuts your DropShip count to 600 (or 2,400 Mules). That's looking a lot more manageable. 800 JumpShips and 2,400 Mules isn't too extreme for the 2780s.
It does, however, indicate how massive losses were in the Succession Wars. The Lyrans couldn't replicate the feat on a smaller scale for the El Diablo incident.