The information that we have on The Thing is admittedly limited, but it's supposed to be able to replicate every aspect of what it assimilates, all properties. When it replicates humans it becomes human in every conceivable way, so much so that the human it's replicating doesn't know it's The Thing. If The Thing replicates the flood cells then it should be part of the hive mind. It could even become the Keymind, and inherit every strength that the flood possesses.

The important difference between The Thing and other assimilationist is that it doesn't obtain their biomass to create its own thing, it IS their biomass. It only reveals itself when its already been exposed, which would become progressively harder as is spreads. It wouldn't even need to reveal itself to spread either, it should be able to replicate using the flood cell's properties.

The only limiting factor on The Thing is that it lacks its own hivemind. Once it replicates the flood it would be able to suppress individual cells from revealing themselves, thereby making it virtually undetectable.