There's been a ton of interesting stuff written about Lincoln, but I haven't seen much about the depiction of patronage as a key lever of presidential power. A good deal of screen time is taken up with the notion that lame duck members of congress who were defeated in November might be persuaded to vote Lincoln's way in exchange for various random civil service jobs—tax collector here, port inspector there—a practice that violates a whole long trajectory of modern civil service rules.
The beginning of the anti-patronage lineage, incidentally, is the Pendleton Civil Service Reform Act sponsored by none other than George Pendleton of Ohio—portrayed in the film as an odious white supremacist and leading opponent of the 13th Amendment.
But what I think is striking looking back from the modern day isn't so much the decline in the president's patronage powers as the decline in the relative desirability of those kind of jobs. A modern-day ex-congressman has much better employment opportunities than becoming a mid-level bureaucrat in western Ohio. He can go work for a lobbying firm. He can run a trade association. Maybe get a job at an ideological group of some kind. You might want an executive appointment from a same-party president—Spencer Abraham became Energy Secretary after Debbie Stabenow beat him in the 2000 election—but even so that tends to serve as a stepping stone into more lucrative things. When Abraham left the government he became a Hoover Institution fellow, launched an "international strategic consulting" firm, and then became chairman of the US subsidiary of a French energy company.
Long story short, these days the federal civil service has lots of solid middle class jobs but there's nothing on offer that would tempt an outgoing member of congress the way the quasi-private sector can.