We report Magnetospheric Multiscale observations of reconnection in a thin current sheet at the interface of interlinked flux tubes carried by converging reconnection jets at Earth's magnetopause. The ion skin depth‐scale width of the interface current sheet and the non‐frozen‐in ions indicate that Magnetospheric Multiscale crossed the reconnection layer near the X‐line, through the ion diffusion region. Significant pileup of the reconnecting component of the magnetic field in this and three other events on approach to the interface current sheet was accompanied by an increase in magnetic shear and decrease in Δβ, leading to conditions favorable for reconnection at the interface current sheet. The pileup also led to enhanced available magnetic energy per particle and strong electron heating. The observations shed light on the evolution and energy release in 3‐D systems with multiple reconnection sites. Plain Language Summary The Earth and the solar wind magnetic fields interconnect through a process called magnetic reconnection. The newly reconnected magnetic field lines are strongly bent and accelerate particles, similar to a rubber band in a slingshot. In this paper we have used observations from NASA's Magnetospheric MultiScale spacecraft to investigate what happens when two of these slingshot‐like magnetic field lines move toward each other and get tangled up. We found that the two bent magnetic field lines tend to orient themselves perpendicular to each other as they become interlinked and stretched, similar to what rubber bands would do. This reorientation allows the interlinked magnetic fields to reconnect again, releasing part of the built‐up magnetic energy as strong electron heating. The results are important because they show how interlinked magnetic fields, which occur in many solar and astrophysics contexts, reconnect and produce enhanced electron heating, something that was not understood before. Key Points Magnetic flux pileup observed upstream of reconnecting current sheet at the interface of converging reconnection jets Magnetic flux pileup was accompanied by increase in magnetic shear and decrease in Δβ, leading to conditions favorable for reconnection Magnetic flux pileup leads to enhanced available magnetic energy per particle and strong electron heating