For most grounds of inadmissibility, most nonimmigrants are actually entitled to a hearing before an immigration judge before being removed from the US. See 8 CFR 235.3(b)(3):
(3) Additional charges of inadmissibility. In the expedited removal process, the Service may not charge an alien with any additional grounds of inadmissibility other than section 212(a)(6)(C) or 212(a)(7) of the Act. If an alien appears to be inadmissible under other grounds contained in section 212(a) of the Act, and if the Service wishes to pursue such additional grounds of inadmissibility, the alien shall be detained and referred for a removal hearing before an immigration judge pursuant to sections 235(b)(2) and 240 of the Act for inquiry into all charges. Once the alien is in removal proceedings under section 240 of the Act, the Service is not precluded from lodging additional charges against the alien. Nothing in this paragraph shall preclude the Service from pursuing such additional grounds of inadmissibility against the alien in any subsequent attempt to reenter the United States, provided the additional grounds of inadmissibility still exist.
However, those who apply for admission under the visa waiver program are not entitled to such a hearing. See 8 CFR 217.4(a)(1):
§ 217.4 Inadmissibility and deportability.
(a) Determinations of inadmissibility.
(1) An alien who applies for admission under the provisions of section 217 of the Act, who is determined by an immigration officer not to be eligible for admission under that section or to be inadmissible to the United States under one or more of the grounds of inadmissibility listed in section 212 of the Act (other than for lack of a visa), or who is in possession of and presents fraudulent or counterfeit travel documents, will be refused admission into the United States and removed. Such refusal and removal shall be made at the level of the port director or officer-in-charge, or an officer acting in that capacity, and shall be effected without referral of the alien to an immigration judge for further inquiry, examination, or hearing, except that an alien who presents himself or herself as an applicant for admission under section 217 of the Act and applies for asylum in the United States must be issued a Form I-863, Notice of Referral to Immigration Judge, for a proceeding in accordance with 8 CFR 208.2(c)(1) and (c)(2).
Note that "at the level of the port director or officer-in-charge, or an officer acting in that capacity" means that the supervisor must already have approved the refusal of admission, so asking to speak to the supervisor may be of limited use.