Source: https://www.ncsl.org/redistricting-and-census/redistricting-criteria
Congressional districts do not have to be contiguous in Arkansas, Connecticut, Illinois, Indiana, Maryland, Massachusetts, New Hampshire, New Jersey, Tennessee, Texas, and Wisconsin. There appears to be no federal statute or court case specifying that Congressional districts be contiguous.
Theoretically, there is nothing stopping any of these state legislatures from engineering Congressional maps that feature non-contiguous districts.
The 1910 Apportionment Act required congressional districts to be "contiguous and compact" and as "nearly as practicable an equal number of inhabitants".
Congress did not apportion after the 1920 Census. When they finally got around to apportioning before the 1930 Census, they left out these provisions as well as the transitional provisions for states with a change in representation. The SCOTUS ruled that this omission was deliberate and that states that had lost representation were not required to redistrict. In effect the apportionment law only had effect until another law was passed superseding it.
After the 1940 Census Congress provided for a permanent apportionment method which would be done automatically. They reinstated the transitional redistricting requirements, but did not enact any requirements for the districts.
In
Wesberry vs. Sanders, the SCOTUS decreed that districts be as "nearly as practicable an equal number of inhabitants." Since it could find no support for this rule in statute except where it had been omitted, they claimed to have found it in the apportionment and elections clause that requires representatives to be chosen by the people and the apportionment be based on population (See dissent by Justice Harlan).
Subsequently, federal district courts were threatening to impose at-large elections as a remedy (as opposed to a federal court meddling in state elections). Congress hastily passed the requirement to require immediate districting.
Over time, the subjective "as nearly as practicable" has been interpreted ever more rigidly (until the West Virginia case after the 2010 Census). In a case in Pennsylvania (Bandemeer?) a court had ruled that a tiny inequality was evidence of a political gerrymander. The SCOTUS ruled that political gerrymanders were justiciable. By the time the case got back to the district court Pennsylvania had fixed the population deviation.
When someone proposes a map they will typically claim that the districts follow "traditional" redistricting criteria such as compactness and contiguity, respect for political subdivisions, and population equality. The challengers in a law suit will point out the strange surgical instruments in the basement of map-drawers.
If a non-contiguous district was drawn, the plaintiffs in the lawsuit will infer motive from that.
