An expiring passport should be renewed before travel is booked rather than tested at a port of entry, because document validity can bear on the period authorized. Do not assume a full period will be granted on a short-dated passport. Renewal is also the step most often started last, once the offer, the credentials and the travel dates are already settled, and that ordering is exactly what turns a routine errand into an emergency.
Renew first, then set the dates
Processing times vary by authority and season, so start the renewal while the employer letter and credential documents are still being prepared. If someone has already been admitted on a short-dated passport, ask counsel about the sequence for renewing and for any extension, and check the admission record so the actual end date is known rather than assumed. Work backwards from the condition that must hold at entry rather than forwards from today.
The passport must be in hand before travel is booked; the letter and credential evidence must match the passport that will actually be presented; the admission record must be checked once entry has happened. Each step depends on the one before it, so a renewal begun late does not merely delay itself, it invalidates work already done under the old document number. Where a start date has been promised, say plainly that it rests on a renewal nobody controls.