One possibility is to find an asteroid that moves between earth and mars orbits, and hitch a ride on it for the bulk of the journey. The asteroid's mass can serve as shielding, and also as a source of raw materials.
Mars's moon Deimos might be the best candidate -- delta-v between Deimos and Earth-Moon L4/L5 is about 3800m/s [1], and there's no need to move the bulk of your shielding lower into the gravity well on either end.
In this scenario, you'd want to use something like an Aldrin Cycler [2] to provide radiation shielded transport between Earth and Mars. This isn't exactly the same thing as hitching a ride on an asteroid, but it's the same basic principle.
You'd definitely need nuclear electric or maybe solar electric propulsion to make such a scheme work, though.
[1] https://en.wikipedia.org/wiki/Delta-v#Delta-vs_around_the_So...
In fact, a large, heavy one can be created or moved into such an orbit, then Mars/Earth journeys only need small, light ships to dock with it at each end.