Electric field between parallel plates

I’m a little confused between two approaches to this problem:

First approach (the one that makes sense to me):
Choose a Gaussian surface over each plate. We have 2EA=σA/Eo so E=σ/2Eo for the positively charged plate and likewise for the negative plate. Within the capacitor, the fields superpose giving E=σ/Eo whilst outside they cancel.

Second approach:
Consider only one plate. We now seem to assume no field passes out one side of the plates, so that EA=σA/Eoand E=σ/Eo. We then don’t bother superposing fields at all.

I don’t understand the second method – surely we’re doing two things wrong:
Ignoring one side of the plate
Not superposing the fields
and these two wrongs happen to make a right.

Can somebody explain this to me please, thanks 🙂

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