Yes, E0 would be nonzero in *EOS_JWL when modeling a high explosive. Values may be hard to come by owing to the sensitive nature of modeling HE.
A fairly common application of E0 in *EOS is to initialize atmospheric pressure in air using *EOS_LINEAR_POLYNOMIAL. Look for the notes on air in the LS-DYNA Aerospace Working Group MGD (Modeling Guidelines Document). Use Google to locate this document.