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Meant to add that my response above was based on US Constitution and quick reading of the exception states to gubernatorial appointment. SD isn't on that list for some reason, but their statutes do provide for a special election at the time of the next general election. SD is not one of the few states that require the governor to appoint someone of the previous incumbent's party, btw.

South Dakota Codified Laws

12-11-5. Special election to fill senate vacancy. The special election to fill the vacancy of a senator shall be held at the same time as the next general election. The general election laws shall apply unless inconsistent with this chapter.

Source: SL 1979, ch 99, § 2.

The US Constitutional difference between replacement of a House member & replacement of a Senator is that a special election must be held in the case of a House member, but is left to the legislature of the state to determine how to replace in the case of a Senator.