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Diverse chemical, energy, environmental, and industrial processes involve the flow of polymer solutions in porous media. The accumulation and dissipation of elastic stresses as the polymers are transported through the tortuous, confined pore space can lead to the development of an elastic flow instability above a threshold flow rate, producing a transition from steady to unsteady flow characterized by strong spatiotemporal fluctuations, despite the low Reynolds number (Re≪1). Furthermore, in 1D ordered arrays of pore constrictions, this unsteady flow can undergo a second transition to multistability, where distinct pores simultaneously exhibit distinct unsteady flow states. Here, we examine how this transition to multistability is influenced by fluid rheology. Through experiments using diverse polymer solutions having systematic variations in fluid shear-thinning or elasticity, in pore constriction arrays of varying geometries, we show that the onset of multistability can be described using a single dimensionless parameter, given sufficient fluid elasticity. This parameter, the streamwise Deborah number, compares the stress relaxation time of the polymer solution to the time required for the fluid to be advected between pore constrictions. Our work thus helps to deepen understanding of the influence of fluid rheology on elastic instabilities, helping to establish guidelines for the rational design of polymeric fluids with desirable flow behaviors.