Do the property assessments in your area represent fair market value (FMV)?  This varies greatly across the country and even within my own state.

A property in Shenandoah County, VA was assessed at $180,500.  It was appraised for $180,000.  Assessments in this area do fairly represent market value.

A property in Fairfax County, VA was assessed at $591,000.  It was appraised for $750,000.  Assessments in this area do not fairly represent market value.

If you paid $350K and your assessment when you placed the property in service was $287K, probably one of two things happened.  Either the assessments do not fairly represent market value in your area.  Or the value of the property declined during the time you lived in it.  IMO neither is more likely that the other, it depends on timing and market conditions.  You are in a better position than anyone here to determine which is more likely.

When you placed the property in service you may have been relying on this tidbit of tax law, explained in IRS Publication 527:

https://www.irs.gov/pub/irs-pdf/p527.pdf

Jump to page 15 under the heading "Basis of Property Changed to Rental Use".

"When you change property you held for per-
sonal use to rental use (for example, you rent
your former home), the basis for depreciation
will be the lesser of the fair market value or ad-
justed basis on the date of conversion"

You may have taken the position that the FMV of the property when you placed it in service as a rental was lower than your cost basis and used the assessment value as an approximation of fair market value.  This could be correct depending on your facts and circumstances.

Rick