President Macky Sall, whose term officially runs out on April 2, announced a postponement of the February 25 vote just hours before official campaigning was due to begin.
The move plunged Senegal into its worst political turmoil in decades.
Last week, the top constitutional body overturned the decision.
However, it has left open the question of whether the election will be organized before Sall leaves office, or if the ballot will feature the same candidates.
Fifteen of the original contenders signed a letter, seen by AFP on Monday, stating that "the new polling date and the date for the handover between the president and his successor must be held no later than April 2."
They added that the list of 20 names approved in January should remain the same.
Two of the signatories confirmed the letter's authenticity to AFP.
Sall has said he intends to respect the Constitutional Council's ruling and will "carry out the consultations necessary to organize the presidential election as soon as possible."
The letter's signatories expressed "bitterness that, since the decision of the Constitutional Council, no action has been taken by the authorities to implement it."
The document was signed by some of the leading contenders, including detained anti-establishment candidate Bassirou Diomaye Faye and former Dakar mayor Khalifa Sall, who is not related to the president.
It was not signed by Prime Minister Amadou Ba, who is the presidential camp's own candidate.