Yes so my suggestion is to slow start the input (Vin) of the lm2596 and use the low voltage lockout to make the lm2596 turn on at ~2.5V. Then the voltage will slowly continue to rise until the slow start has fully turned on.
As shown in the datasheet
http://www.ti.com/lit/ds/symlink/lm2596.pdf :
UNDERVOLTAGE LOCKOUT
Some applications require the regulator to remain off until the input voltage reaches a predetermined voltage. An
undervoltage lockout feature applied to a buck regulator is shown in Figure 34, while Figure 35 and Figure 36
applies the same feature to an inverting circuit. The circuit in Figure 35 features a constant threshold voltage for
turn on and turn off (zener voltage plus approximately one volt). If hysteresis is needed, the circuit in Figure 36
has a turn ON voltage which is different than the turn OFF voltage. The amount of hysteresis is approximately
equal to the value of the output voltage. If zener voltages greater than 25V are used, an additional 47 k? resistor
is needed from the ON /OFF pin to the ground pin to stay within the 25V maximum limit of the ON /OFF pin.
So try a 1.8-2V zener, or even up 4V. Then when it turns on you won't see the output of the lm2596 jumping any higher than the voltage on the Vin side.