The term “residual hearsay exception” refers to exceptions to the hearsay rule that are not specifically set forth in the Federal/State Rules of Evidence, but can be used to admit hearsay statements provided that those statements satisfy the general trustworthiness requirements of the hearsay rule.
Idaho v. Wright, 497 U.S. 805 (1990)
The residual hearsay exception . . . accommodates ad hoc instances in which statements not otherwise falling within a recognized hearsay exception might nevertheless be sufficiently reliable to be admissible at trial.
Federal Rule of Evidence 807 – Residual Exception
(a) In General. Under the following circumstances, a hearsay statement is not excluded by the rule against hearsay even if the statement is not specifically covered by a hearsay exception in Rule 803 or 804:
(1) the statement has equivalent circumstantial guarantees of trustworthiness;
(2) it is offered as evidence of a material fact;
(3) it is more probative on the point for which it is offered than any other evidence that the proponent can obtain through reasonable efforts; and
(4) admitting it will best serve the purposes of these rules and the interests of justice.
(b) Notice. The statement is admissible only if, before the trial or hearing, the proponent gives an adverse party reasonable notice of the intent to offer the statement and its particulars, including the declarant’s name and address, so that the party has a fair opportunity to meet it.