I have never seen a property assessed higher than the sales price of any property in SC. Usually, the new point of sale value used by the County assessor is what's on the deed they receive to record when the property closes. From there on, the assessed value typically lags behind the fair market value. In SC, you can not be assessed more than a 15% increase YOY from the previous year's assessed value.
Also, the only distinction in tax rates are 4% (owner occupied homes) and 6% (commercial rate). The actual tax rate is not 4% or 6%. That's the starting value before millage and credits are applied. For commercial properties, the tax bill typically ends up around 1% of the sale price. That's a generic rule, but relatively close. The 6% rate applies to everything from a single family rental to an industrial warehouse. The county knows the difference because a homeowner must submit an application for the 4% rate in order to get it. That 4% rate is amongst the lowest property tax rate in the country. The 6% rate is roughly 2-3x the 4% rate, so it is a significant jump when you do not occupy the property.
You can also get a tax exemption to reduce the new tax bill by 25% if there's a significant enough increase in value from the previous assessed value and the new point of sale value. That law is called the ATI tax exemption. It's a little vague, but can be extremely helpful if you can qualify for it.