Rincón won the 2021 US Open boys' singles title, defeating Shang Juncheng in the final.

[2] He also reached the 2021 Wimbledon Boys' Doubles final alongside Abedallah Shelbayh, but was defeated by Edas Butvilas and Alejandro Manzanera Pertusa.

[3] Rincón won his first ATP Challenger doubles title at the 2023 Murcia Open, partnering Abedallah Shelbayh after they received entry into the draw as alternates.

[5] He qualified for the main draw at the 2023 Geneva Open but lost in the first round to Roberto Carballes Baena in three sets.

[6] Rincón received a wildcard for the 2024 Barcelona Open Banc Sabadell but lost to Facundo Díaz Acosta.