Step 1: The clue word is "best demonstrated." In pyloric stenosis the diagnostic finding is the palpable pyloric tumour, but a thickened pylorus the size of an olive is hard to feel in an unsettled, dehydrated baby with a rigid abdominal wall.
Step 2: The trick used by clinicians is the test feed. While the mother or nurse offers a bottle, the relaxed and contented infant allows deep palpation; the gastric distension also pushes the pylorus forward. Under these conditions both the firm olive and the left-to-right peristaltic waves of gastric outlet obstruction become apparent. So the answer is during feeding.
Step 3: Rejecting the rest: plain epigastric palpation alone is unreliable for the reasons above; the left hypochondrium and right iliac fossa do not correspond to the pyloric region, which sits in the epigastrium just right of the midline. Definitive confirmation is by ultrasound, and treatment is Ramstedt pyloromyotomy after correcting the hypochloraemic alkalosis.
\[\boxed{\text{During feeding}}\]