I don't think there is need for any special election for president. Johnson's replacement simply serves out the term. Foster unlike Wade wasn't a fireband (apparently he wanted a hugly strong presidency and equal rights for women, two big no-nos back then)
Well, there is such a need if Section 10 of the Presidential Succession Act of 1792 (then still in effect) is followed:
Sec. 9.
And be it further enacted, That in case of removal, death, resignation or inability both of the President and Vice President of the United States, the President of the Senate pro tempore, and in case there shall be no President of the Senate, then the Speaker of the House of Representatives, for the time being shall act as President of the United States until the disability be removed or a President shall be elected.
Sec. 10.
And be it further enacted, That whenever the offices of President and Vice President shall both become vacant, the Secretary of State shall forthwith cause a notification thereof to be made to the executive of every state, and shall also cause the same to be published in at least one of the newspapers printed in each state, specifying that electors of the President of the United States shall be appointed or chosen in the several states within thirty-four days preceding the first Wednesday in December then next ensuing:
Provided, There shall be the space of two months between the date of such notification and the said first Wednesday in December, but if there shall not be the space of two months between the date of such notification and the first Wednesday in December; and if the term for which the President and Vice President last in office were elected shall not expire on the third day of March next ensuing, then the Secretary of State shall specify in the notification that the electors shall be appointed or chosen within thirty-four days preceding the first Wednesday in December in the year next ensuing, within which time the electors shall accordingly be appointed or chosen, and the electors shall meet and give their votes on the said first Wednesday in December, and the proceedings and duties of the said electors and others shall be pursuant to the directions prescribed in this act.
https://en.wikisource.org/wiki/Unit...e/Volume_1/2nd_Congress/1st_Session/Chapter_8
The language of Section 10 certainly looks mandatory: "the Secretary *shall*," "electors ...*shall* be appointed or chosen...within thirty-four days preceding the first Wednesday in December then next ensuing.." etc. Not "may" but "shall."