The HEDI/Sprint/Spartan mash-up is representative of the Phase III HEDI test vehicles, the US Army got as far checking Spartan motors to see whether or not they had not degraded beyond the point of usability for the test programme.
HEDI was intended for very long range endoatmospheric shots, in part to allow 'shoot-look-shoot' engagements. HEDI was to have a much higher final closing speed than even Sprint, 19,000ft/sec versus 10,000ft/sec in Sprint. This would have required a larger booster which is probably what is depicted in the first picture in this thread. It also explains why the ALPS proposals that used HEDI placed it in Spartan silos rather than Sprint silos - because it needed space for an absolute beast of a booster.
There was also a plan for a HEDI follow-on, the HEDI E2I, this would have had a dual-wave (MWIR and LWIR) seeker and a further expanded engagement envelope potentially including low exoatmospheric targets.
Sources:
Subcommittee of the Senate Appropriations Committee - Hearings on the DoD Appropriations for FY86
Subcommittee of the House Appropriations Committee - Hearings on the DoD Appropriations for FY88
Weapon Systems of the United States Army, 1989, US Army
House Committee on Armed Services - Hearings on National Defense Authorization Act for FY91
