If Hindenburg doesn't run it's possible that Reinhard Scheer would step in as the conservative candidate.
Two problems with Scheer: First, given the narrowness with which Hindenburg won, I doubt that Scheer--a war hero, yes, but not really with Hindenburg's prestige--could have won in the second round even if all the right-of-center parties had united around him. Second, and most important, Stresemann--who would brook no rivals within the DVP-- would almost certainly have opposed Scheer. As Henry Ashby Turner writes in
Stresemann and the Politics of the Weimar Republic, pp. 230-1:
"Curtius, who remained as Minister of Economics in the new cabinet, later complained in his memoirs about the fact that he had received less support from Stresemann in his bid for the chancellorship than Marx. His conclusion was that, as Foreign Minister, Stresemann feared he would be a more forceful Chancellor than Marx and would thus want a larger voice in the formation and execution of foreign policy. Another possible explanation, however, could be that Stresemann was not pleased at the prospect of sharing the limelight within the DVP with a Chancellor or another ex-Chancellor. According to Hans von Raumer, a DVP deputy from 1920 until 1930, he was constantly apprehensive about possible rivals. On at least two earlier occasions he had shown himself extremely sensitive to challenges to his position of preeminence within the party. The first of these occurred in 1922, when Admiral Reinhard Scheer, a hero of the war, joined the DVP and opened an obvious bid for recognition with a series of speeches in various parts of the country. For a time Stresemann tolerated the Admiral's actions. But his patience quickly faded when Scheer aligned himself with the extreme left wing of the DVP and began criticizing him for not making more progress toward cooperation with the SPD. Reacting angrily, he brought the Admiral's bid for power to an abrupt end by securing a motion of censure against him from the Reichstag delegation. [In a footnote, Turner adds: "Scheer remained in the party for a time, but was completely without influence."] The second incident occurred in 1923 after the end of his own chancellorship, when President Ebert called upon Siegfried von Kardoff of the DVP to form a government. Kardoff accepted but had to withdraw at once in large part because, to his surprise, Stresemann refused to remain on as his Foreign Minister--although Stresemann then promptly agreed to do just that under the Centrist Marx. These two incidents, when added to Curtius' experience, would seem to substantiate Raumer's observation and to indicate that Stresemann preferred to remain as the DVP's one nationally prominent figure, even at the cost of sacrificing importamt increases in the party's strength and influence."
https://books.google.com/books?id=n0rWCgAAQBAJ&pg=PA230
https://books.google.com/books?id=n0rWCgAAQBAJ&pg=PA231
(If Scheer actually did urge the DVP to cooperate more with the SPD, I could see many German conservatives other than Stresemann who might have a problem with him...)