LXE-HX2i decrease battery voltage, step-down converter
Drop to 3.3V is ok by the 3.3V version of LM2596. The 5V version needs input=7V or more. It's possible that your circuit does not work when the battery is only half full or nearly empty. See the specs from the original datasheet: http://www.ti.com/lit/ds/symlink/lm2596.pdf