Before travel, before any inspection or filing, and before the first day of work. A mismatch discovered at that point is far harder to correct than one addressed while the employer's records are already being updated.
Use the employer's own transition window
Companies changing their legal name are already reissuing contracts, payroll records and letterhead, so the corrected support letter costs little effort if it is requested during that work. Waiting until the documents are needed means asking a signatory to produce a specific letter under time pressure. If the entity changed rather than renamed, the earlier counsel is involved, the more options remain open.
Put a date on each dependency and work backwards. The registry extract governs everything else, so request it first; the reissued letter can be drafted in parallel but should not be finalised until the extract confirms which name belongs on it. Build in the credential documents too, since a certified translation or an evaluation can take longer than the corporate paperwork it accompanies.
Hypothetical example: a lens-materials chemist is due to start at the beginning of a quarter, and the company’s name change takes effect two weeks earlier; requesting the letter inside that window costs nothing extra, while requesting it afterwards means competing with the finance team’s own year-end work. If the entity actually changed, allow time for counsel to choose between a request at a port of entry and a petition, because that choice moves the calendar more than any single document.