Antihydrogen, the bound state of an antiproton and a positron, is of interest for use in precision tests of natureʼs fundamental symmetries. Antihydrogen formed by carefully merging cold plasmas of positrons and antiprotons has recently been trapped in magnetic traps. The efficiency of trapping is strongly dependent on the temperature of the nascent antihydrogen, which, to be trapped, must have a kinetic energy less than the trap depth of ∼ k 0.5 K B . In the conditions in the ALPHA experiment, the antihydrogen temperature seems dominated by the temperature of the positron plasma used for the synthesis. Cold positrons are therefore of paramount interest in that experiment. In this paper, we propose an alternative route to make ultra-cold positrons for enhanced antihydrogen trapping. We investigate theoretically how to extend previously successful sympathetic cooling of positrons by laser-cooled positive ions to be used for antihydrogen trapping. Using simulations, we investigate the effectiveness of such cooling in conditions similar to those in ALPHA, and discuss how the formation process and the nascent antihydrogen may be influenced by the presence of positive ions. We argue that this technique is a viable alternative to methods such as evaporative and adiabatic cooling, and may overcome limitations faced by these. Ultra-cold positrons, once available, may also be of interest for a range of other applications.