Would Hamlin have Lincln's political finesse in dealing with the various political factions in the North (and Europe)? He was one of the Radicals and may have stepped in it badly in dealing with his generals, whom even Lincoln found exasperating. An overreaction on the Trent Affair could easily have greatly complicated the path to victory, or even resulted in a loss to McClellan in 1864.
The immediate risk is Hamlin pushing his abolitionist agenda too soon and too openly. Pissing off Maryland leaves Washington D.C. surrounded by enemy territory, pissing off even a single border state stacks so much the situation in favour of the confederacy to make the secession almost a open and shut case.
Further down the road the issues become handling the Trent affair calmly, restraining Seward aka as the "we will wrap the world in flames" guy

and dealing with generals, especially with the nearly mutinous McClellan.
We do really need some good insight into Hamlin psychology in order to assess the potential of this POD.
If you want to pursue this, one possible scenario is for L to die of mercury poisoning. He was taking something called Blue Mass in 1861 but quit it bc he said it made him cross. It was something of a patent medicine then with he was unknowingly slowly poisoning himself.
Or just even fall in the bathroom. Statistics show that the cozy bathroom is the most dangerous room in the house. Don't forget that we are leaving in a pre-sulphamidics/pre-antibiotics era: even a scratch can turn into a deadly infection.
Another cause of death are riding accidents as King William III is glad to confirm (dying by falling from a horse who tripped into a mole burrow? ASB, isn't it?

).
Idle meditation: Lincoln dies soon after his election by a somewhat freakish accident. Could such event be read by more religious people (even in the north) as a sign of divine displeasure with the whole abolitionist idea?