Step 1: A diffuse bilateral shadowing pattern tells us the lung interstitium is involved, but a flat two-dimensional radiograph blurs the fine detail needed to classify it.
Step 2: High-resolution CT solves this by giving thin-section images that reveal the exact texture and zonal spread of the disease, such as ground-glass change or honeycombing. For this reason it is performed in almost every case of diffuse parenchymal disease and is the appropriate next test.
Step 3: Looking at sputum or starting antibiotics assumes infection, which a diffuse interstitial pattern does not support. Bronchoscopy is more invasive and is reserved for after imaging has guided where to sample.
Step 4: Therefore the immediate follow-up to the chest X-ray is HRCT.
\[\boxed{\text{CT (HRCT) scan}}\]