I neglected to mention that the examples I offered are not re-letterings, but total re-paints with new lettering, as all of the cars shown were modified to some degree.
I have relatively few cars with factory lettering, but this Walthers boxcar has factory paint and mostly factory lettering...
The re-weigh data is dry transfer lettering, and appears to be applied over an area where the original lettering was scraped off as described. I either paint that area or, if it's in good condition, mask it before weathering. After the initial weathering, the masking is removed, revealing what looks like new paint, and the re-weigh data is applied.
The re-pack info, on the small black patch on the right-hand side of the car, is a Champ decal, while the black background is either painted directly on the car or is a piece of clear decal, painted black as part of a larger sheet, then cut out and applied prior to the repack info.
After the data has been updated in this manner, many of these cars get some additional weathering, depending on the date shown in the re-weigh data.
In my layout's late '30s era, most cars required re-weighing every 30 months, but that time period varied over the years, so you'll need to check the requirements for your chosen era.
All of my photos are in photobucket, and clicking on the picture will take you there, where a full-screen view is available.