We study the dependence of Saturn's magnetospheric magnetic field structure on the interplanetary magnetic field (IMF), together with the corresponding variations of the open-closed field line boundary in the ionosphere. Specifically we investigate the interval from 8 to 30 January 2004, when UV images of Saturn's southern aurora were obtained by the Hubble Space Telescope (HST), and simultaneous interplanetary measurements were provided by the Cassini spacecraft located near the ecliptic ~0.2 AU upstream of Saturn and ~0.5 AU off the planet-Sun line towards dawn. Using the paraboloid model of Saturn's magnetosphere, we calculate the magnetospheric magnetic field structure for several values of the IMF vector representative of interplanetary compression regions. Variations in the magnetic structure lead to different shapes and areas of the open field line region in the ionosphere. Comparison with the HST auroral images shows that the area of the computed open flux region is generally comparable to that enclosed by the auroral oval, and sometimes agrees in detail with its poleward boundary, though more typically being displaced by a few degrees in the tailward direction.