While numerous fundamental studies have sought to elucidate the effect of a parent homopolymer on the morphological characteristics and mechanical properties of microphase-ordered block copolymer blends, few comparable efforts have extended such studies to concentrated copolymer solutions in the presence of a low-molar-mass block-selective solvent. In this work, we investigate the microstructures that form in blends of a poly(styrene-block-isoprene-block-styrene) (SIS) triblock copolymer with a midblock-selective aliphatic mineral oil. To discern the influence of midblock/oil compatibility on blend morphology and properties, identical blends with a poly[styrene-block-(ethylene-alt-propylene)-block-styrene] (SEPS) copolymer, the hydrogenated variant of the SIS copolymer, have likewise been examined. The saturated midblock of the SEPS copolymer is responsible for the observed shifts in morphology stability limits and higher dynamic elastic sheer moduli relative to the SIS analogue. These results reveal that the morphologies and properties of such triblock copolymer/oil blends are sensitive to the chemical/statistical nature of the copolymer midblock and may be judiciously tailored to satisfy application-specific requirements.