S.D. Codified Laws § 34-10-10
The boundaries of any public hospital district may be altered and outlying districts annexed from territory contiguous thereto in the following manner: a petition signed by sixty percent of the resident electors within the territory proposed to be annexed, designating the boundaries of such contiguous territory proposed to be annexed and asking that it be annexed to said public hospital district, shall be presented to the board of trustees. At the first regular meeting after the presentation of said petition, said board shall cause notice of said petition to be published in two successive issues of an official newspaper published in each county in which said district is situated prior to the date fixed by the board for a hearing of said petition, which date shall not be less than four weeks after the filing of such petition.
Upon the date fixed for such hearing or continuance thereof, said board shall take up and consider said petition and any objections which may be filed to the inclusion of any additional area or territory in said district. Said board shall have the power by order entered on its minutes to grant or deny said petition, and by order entered on its minutes to alter the boundaries of said hospital district and to annex thereto said area or territory described in said petition. This territory shall become and be a part of such public hospital district and shall be subject to the tax authorized by this chapter, together with the preexisting area of said district.
SDCL 34-10-10