You said it right, he did evolve over time to want to extend the right to vote to black Union soldiers. As Louis P. Masur writes in his book Lincoln's Last Speech: Wartime Reconstruction and the Crisis of Reunion, this is a position that Lincoln had held privately since at least mid-1864, when he proposed it to some Louisiana Republicans in private letters.

But, according to historian Eric Foner, that was only ever the extent of it. He writes in his book The Fiery Trial: Abraham Lincoln and American Slavery:

Unlike [U.S. Sen. Charles] Sumner and other Radicals, Lincoln did not see Reconstruction as an opportunity for a sweeping political and social revolution beyond emancipation. He had long made clear his opposition to the confiscation and redistribution of land. He believed, as most Republicans did in April 1865, that voting requirements should be determined by the states. He assumed that political control in the South would pass to white Unionists, reluctant secessionists, and forward-looking former Confederates.

Foner adds, though:

But time and again during the war, Lincoln, after initial opposition, had come to embrace positions first advanced by abolitionists and Radical Republicans.

A few pages earlier, Foner discusses what is often called Lincoln's Last Speech, delivered outside the White House on April 11, 1865, where he gave public support for the right to vote for black soldiers for the first time. He simultaneously acknowledged that some white Americans supported full, universal suffrage for black men:

"It is also unsatisfactory to some that the elective franchise is not given to the colored man. I would myself prefer that it were now conferred on the very intelligent, and on those who serve our cause as soldiers."

As Foner details, earlier in the speech, Lincoln contended that he had submitted his Reconstruction plan to his Cabinet for approval and "every member of the Cabinet fully approved the plan". When submitted to Congress, Lincoln claimed there was "not a single objection to it", nor "from any professed emancipationist".

However, writes Foner, this wasn't exactly true. Salmon P. Chase, former Cabinet member and then-current Chief Justice of the Supreme Court who had attended the meeting, wrote to Lincoln the next day, to say that he actually had protested the plan, on the basis that it did not enfranchise all black men. But he forgave Lincoln in the same letter, because Chase admitted his protest had not been very strong, so as not to come off as "pertinacious".

Even so, Foner writes that Lincoln's more modest plan still represented a powerful shift:

This [the support for soldier enfranchisement in Lincoln's April 11 speech] was a remarkable statement. No American president had publicly endorsed even limited black suffrage. At this time only six northern states allowed black men to vote.

Further, Foner does give some support to the idea that maybe Lincoln would have come around to the idea of universal black male suffrage that many of the Radicals were pushing for. On April 14, Lincoln met with his Cabinet:

Before the meeting, he [Lincoln] showed Attorney General James Speed a letter he had received from Chase urging the enfranchisement of all “loyal citizens” regardless of race when new state governments were formed. Speed thought Lincoln was moving toward the Radical position. “He [never] seemed so near our views,” Speed told Chase the next day. Lincoln now appeared to believe that the immediate problem was the prospect of anarchy in the South. He had directed Secretary of War Stanton to draw up a plan for interim military rule....Since his plan put off the establishment of civilian rule, he “left open” the question of whether blacks should vote. But Stanton’s proposal clearly implied that reliance on white Unionists might not be enough to establish loyal, stable governments. Little discussion ensued, and Lincoln urged his colleagues to devote their attention to “the great question now before us,” on which “we must soon begin to act.” Stanton was directed to redraft his proposal for consideration at the next cabinet meeting.

Alas, Lincoln was killed before that next Cabinet meeting was held.

So, it would seem that Lincoln was at least still open to the idea of universal black male enfranchisement at the time of his death. Nevertheless, as Foner also notes, several other Radicals wrote to each other in critical terms over those last few days of Lincoln's life, criticizing his Reconstruction plan specifically because of its conservative approach to the voting issue.

Masur writes along the same lines, though he believes that Lincoln's words on the subject were more diplomatic in nature, rather than a strongly held personal belief. Masur details how Lincoln had already come to support at least limited suffrage of black men by 1864, but kept it quiet:

Lincoln had good reason not to make his preference [on the voting issue] known. Most Americans viewed extending the vote to blacks as more radical than eliminating slavery.

Foner gives some credence to the idea, relating a story of John Wilkes Booth who is said to have attended Lincoln's Last Speech (slur censored):

One member of the audience, however, thought he understood exactly what Lincoln intended. "That means n****r citizenship," the actor John Wilkes Booth is said to have remarked. Booth and a group of pro Confederate conspirators had been plotting to kidnap the president and demand the release of southern prisoners of war. "Now, by God," Booth supposedly muttered, "I’ll put him through."

The story may be apocryphal, but it first appeared in print shortly after Lincoln's and Booth's deaths, so even if it's not true, it was at least considered plausible for the time, reflecting the thoughts of many white Americans, both North and South.

So, had Lincoln's views evolved by April 1865 enough that it's reasonable to think he would have been a supporter of the 15th Amendment by then, or shortly after? Only the most generous reading of the surviving primary sources would come to that conclusion.

But was it possible that Lincoln may have continued to "evolve" on the issue throughout his term, and come around to the Radical position? It is certainly possible. Lincoln's main concern as the war was coming to its conclusion was how to preserve the peace, and he did seem open to the idea that universal enfranchisement of black men may be necessary in order to make that happen. Then again, in light of the Booth anecdote, Lincoln may also have considered it too risky, and a threat to peace and a reunited country.

Ultimately, it will forever remain a mystery due to Lincoln's unexpected death. Events from then on could have turned out entirely different than they did. All we can say for sure is that Lincoln supported limited suffrage at the time of his death, and had not yet quite ruled out universal suffrage. But all of his writings up until his death reveal that Lincoln believed that limited suffrage, via state laws, was the best path forward.