I've mentioned the possibility of the Arkansas Statehood Centennial half dollar being "graced" with three different designs before, particularly when discussing the Arkansas-Robinson type which resulted from the three-design discussion, but wanted to return to the topic and dig into it a bit more.
In March 1936, Representative David Dickson Terry (D-AR) introduced a bill in the House of Representatives that provided for a change in design for the Arkansas Statehood Centennial half dollar by directing the Treasury/US Mint to create "a series of not more than three different designs to be placed on the reverse side of the 50-cent pieces" already authorized. (Without explicit evidence to the contrary, my working assumption is that the bill called for three new designs in addition to the conjoined portrait design already being struck.)
Upon its introduction, the bill was referred to the House Committee on Coinage, Weights and Measures. The Committee supported one additional design for the Arkansas coin vs. the three that the bill would have created. In its Report, it recommended that the bill be passed with the following amendment "strike out the words 'a series of not more than three different designs' and insert 'one additional design of not less than twenty-five thousand'".
The bill and Committee's recommendation were considered by the House and passed. Minor discussion regarding the number of commemorative bills being presented to Congress occurred. but consideration of the bill was allowed to continue and for it to be passed by Unanimous Consent. The bill was then sent to the Senate for its consideration.
Received in the Senate, it was referred to the Committee on Banking and Currency. The Committee recommended the bill be passed, but with several recommended amendments. The amendments included: capping the mintage for the new design at 50,000 coins (vs. being open-ended within the 500,000 approved for the original design), that striking of the new design be limited to a single US Mint location (i.e., no P/D/S sets) and specifying that the coining authority for the new coin would expire one year after the bill's enactment.
The Whole Senate considered the bill and its recommended amendments and passed it without issue. The bill was then sent back to the House for its concurrence.
The bill with Senate amendments was brought up for consideration in the House with a request for Unanimous Consent. After a very brief discussion - not truly germane to the Arkansas coin bill's consideration - the House concurred in the Senate amendments and passed the bill.
The bill was then engrossed and signed in each chamber and presented to the President for his approval. President Franklin Delano Roosevelt signed the bill into law on June 26, 1936.
Had the original version of the bill passed, collectors would have had multiple new Arkansas designs to pursue. The revised single-design version, however, did at least pave the way for the Arkansas Statehood-Senator Joseph Robinson design type and gave collectors a bonus of one new type to add to their cabinets!
1936 Arkansas Statehood Centennial - Senator Joseph T. Robinson Design Type

For more of my posts about the Arkansas Statehood Centennial half dollars (and other commemorative coins and medals), check out:
Commems Collection.