On an Earth-to-Mars transfer, a hyperbolic trajectory is required to escape from Earth's gravity well, then an elliptic or hyperbolic trajectory in the Sun's sphere of influence is required to transfer from Earth's sphere of influence to that of Mars, etc.
For a journey from Earth to Mars, a hyperbolic trajectory is required to leave Earth's gravitational field. Following that, an elliptical or hyperbolic path within the Sun's sphere of influence is needed to move from Earth's sphere of influence into Mars' sphere of influence, and this pattern continues.