I would wait at least a week or two after moving an established tank to add new fish, and ideally closer to a month. Moving the tank does a few things to disturb its normal equilibrium, so it's just safer to wait until things have settled back down.
It's really important to monitor your water quality after the move. For one, in a lot of cases you're doing a huge water change -- probably 75% or more with a larger tank. If the pH, temperature, or hardness of the new water is dramatically different, this not only directly affects your fish, but your biofilter as well.
You also kick up a lot of detritus from the substrate. All this rotting food, fish waste, bacteria, etc. can easily impact your water chemistry -- it's potentially a lot more than the biofilter can handle right away. And the mechanical filter will have to deal with an above-average amount of, shall we say, fine particulate organic matter.
Also bear in mind that the existing fish have just gone through a highly stressful event. Even if the water chemistry is 100% perfect after the move, they're going to be more prone to health problems for a little bit. The immune system of a healthy fish will normally prevent the latent diseases in an aquarium from actually causing any sickness, but this level of stress can interfere with that. You don't necessarily need to treat the fish as if they're in quarantine after a tank move, but it's good to just watch them carefully over the next few weeks just in case. And any new fish is a potential source of new diseases as well -- it's safer to wait until your fish have settled in, rather than adding this at a point where they're more susceptible than usual.
This is sort of unrelated to the new-fish part of your question, but be extremely careful when you move a tank to get as much water as possible out of it. Water is shockingly heavy (when full, your tank weighs more than a grown man), and though the silicone holding a tank together is extremely strong, the shifting weight of water as it sloshes around can put mechanical stresses on side seams that aren't meant to be load bearing. If it's at all possible, try to take all the water, substrate, and decorations out of a tank when moving it, to prevent damage.