Voltage level detection circuitry
Some low-cost ICs rely on internal POR (power-on-reset) circuitry to start operating. And they may not have a dedicated RESET pin. This kind of POR circuitry usually requires certain Vin ramp to work properly. For example, Microchip states the ramp rate of 50mV/ms for PIC16 familiy, which is about 100 ms ramp time. The TPS54302 has the output ramp time of about 4 ms (10-90%), which should easily fit the bill, unless something else is designed-in. 

If the 4-5 ms for 5-V rail is too slow for the IC in use, you might need to design-in a high-side power switch controlled by a simple RC delay. Or maybe just reducing some unnecessary-high bypass capacitance on board will fix the start-up issue. The details are in your numbers, which you decided to omit from your question.