A catheter system includes a flexible abutment at its distal end. Functionally, the abutment serves to stabilize and align (orient) the distal portion of the catheter as its distal tip is advanced into contact with a tissue surface. Structurally, the flexible abutment is an arrangement of wires that creates a hollow conical shaped frame having two open ends. One end is affixed to the distal end of the catheter. And, the other is presented as a plurality of separated end points that collectively define an annulus at a distal distance from the catheter's tip. For its operation, the flexible abutment is moved from a folded (tube-shaped) configuration, for travel through the vasculature of a patient, to a flared configuration with end points on the frame extended outwardly from the catheter axis, for contact with the tissue surface.