A new minimum-fuel glideslope guidance algorithm for approaching autonomously a target evolving on an elliptic orbit is proposed in this paper. Assuming chemical propulsion, the present work aims at enhancing Hablani's seminal method. Although this reference method is efficient and easy to implement, there is no direct control on the fuel consumption and the guidance error. By identifying some relevant degrees of freedom, a new formulation of the glideslope guidance algorithm is proposed to address these issues. For a fixed-time rendezvous and a given number of maneuvers, the fuel-optimal multipulse glideslope is expressed as a semidefinite programming problem. A solution to this optimization problem provides an impulsive control sequence that guarantees a minimal consumption and makes sure the chaser's trajectory remains included inside a user-defined corridor. Besides, if trajectory constraints are removed, or if a specific direction is considered (V-bar or R-bar), then the formulation is reduced to a simple linear programming problem. Three realistic scenarios of rendezvous are simulated to illustrate the benefit of the proposed methodology.