There is no need to go to the SSA to update your visa status. Your card remains valid, and you have authorization to work.
The only time you MUST have it changed is when your card was issued as "ineligible" to work and then you can.
Some like to have their card updated when they get their Green card, but this is completely optional and not required, since a Green card is still authorization.
I would consider getting a new SSN if one became a US citizen, but, agin, this would be optional.
But there is no need to go to SSA office to change when you go from TN to TD to H1 or H4 or F1 or anything else. It's not a SSA matter once they issue the card.