Stories of oiling fans to make them quieter are OUT OF DATE by two or three decades! Fans in the 80's and 90's were constructed so that you could disassemble them , clean the front and rear bearings, add a small bit of new oil, re-assemble and re-install. Just like you can do with a small desktop fan today. But for a long time now computer fan designs have been quite different, and there is NO way to disassemble without breaking the fan parts, then re-assemble to work properly. Trying to add oil when you cannot disassemble means only that you are pushing oil into places it does NOT belong and making things worse!
Besides, as several people have posted above, the real source of that noise is wear of the bearing causing excess clearance. That does exactly what you say - noise when the fan starts up cold, then quiet after it warms up. Even with old fans, adding oil just was a temporary step until the wear become much worse. The real solution is replacement.