By 1865, Hamlin was a Radical Republican, so domestically he would've been similar to Seward, but Seward wasn't in the line of succession. Next would've been President pro tem Lafayette Foster who would've served as Acting President until the election of 1865. One interesting side effect would've been the decoupling of Presidential and Congressional elections.
What?
How does this work?
The Presidential Succession Act of 1792, which established the President pro tempore as next in line after the Vice President (followed by the Speaker of the House), only provided for these officers to "
act as President of the United States until the disability be removed or a president be elected."
The Act called for a special election to be held in November of the year in which dual vacancies occurred (unless the vacancies occurred after the first Wednesday in October, in which case the election would occur the following year; or unless the vacancies occurred within the last year of the presidential term, in which case the next election would take place as regularly scheduled).
The people elected President & Vice President in such a special election would've served a full 4-year term beginning on March 4 of the next year, but no such election ever took place.