Most people think of the election as a vote for or against secession. It gives the wrong impression that substantial numbers of men accepted Lincoln's election and wished to retain the union of states. That's not correct at all.

After Lincoln's election, citizens debated the wisdom of immediate secession vs. cooperation with other Southern states. Nearly all of those who voted against secession favored action based on a conference of delegates from each state. The proposed conference would be something like the Continental Congress just before the American Revolution. Georgia would leave the union only upon confirmation that many other states would join her.

Note that most of the delegates who opposed immediate secession supported the Confederacy.

Perusing the debates and resolutions offered during Georgia's secession convention should help.
http://docsouth.unc.edu/imls/georgia/georgia.html
For example, here's a resolution by Rufus J. Cochran of Wilkinson County, who voted against secession --

"Mr. Cochran of Wilkinson, laid on the table the following preamble and resolutions which were read."

The aggressions of the people of the Northern States, of the Republic of the United States of America, upon the constitutional rights of the State of Georgia, having been deemed by the people of Georgia sufficient cause to impel her in the maintainance of her rights and honor, to withdraw her connection with the Federal Union, and to resume her sovereignty and independence, justice to herself requires that she should demand her proportional part of the public property, and intending to preserve untarnished, her honor, in the discharge of every moral obligation, and in good faith to the creditors of the General Government, contracted with while Georgia was a member thereof, urge her to a guarantee of the payment of her pro rata part of the public debt of the United States, existing at the time of the act of secession of this State from the Union.

Be it therefore resolved, That Georgia will demand, and enforce her rights, to her proportion of the public property, held by the General Government at the time of the dissolution of the partnership.

Resolved further, That Georgia will assume and guarantee the payment of her pro rata part of the public debt of the United States, existing at the time of her secession from the Union.

Resolved further, That the foregoing resolutions be communicated to the Governor, with the request that he lay them before the Legislature upon its re-assembling, with the recommendation of this Convention that the Legislature take such action as may be necessary to carry the same into effect.

Resolved further, That this Convention of the people of Georgia, respectfully call the attention of our sister seceding States to this subject, and ask their co-operation in the policy, and that the Governor of Georgia be requested to communicate the same to the respective Governor's thereof.

Read also,

"Mr. Nisbet, offered the following preamble and resolution which were taken up and read:"

Whereas, the lack of unanimity in the action of this Convention, in passage of the Ordinance of Secession, indicates a difference of opinion amongst the members of the Convention, not so much as the rights which Georgia claims, or the wrongs of which she complains, as to the remedy and its application before a resort to other means of redress:

And whereas, it is desirable to give expression to that intention which really exists among all the members of this Convention, to sustain the State in the course of action which she has pronounced to be proper for the occasion, therefore:

Resolved, That all members of this Convention, including those who voted against the said ordinance, as well as those who voted for it, will sign the same as a pledge of unanimous determination of this Convention to sustain and defend the State, in this her chosen remedy, with all its responsibilities and consequences, without regard to individual approval or disapproval of its adoption.

"Mr. Nisbet then moved their adoption, which motion prevailed."