Who's to say that they didn't say, "our offer is 4.2 million the first year, 3.8 million for a second year." KT didn't immediately sign, in order to see if anyone else would offer him MORE salary. This would be in order for him to go back to the Spurs to say, "I have an offer of 5.58 million." In which case, the Spurs would counter-offer 6.
The point being, you're assuming that it was the Spurs not offering him a contract on the first day of Free Agency that was so risky. There was no risk. The risk being the Spurs may have to pay more than they would like (to the tune of 6 million, rather than the 4.2 million they paid). KT is the one looking for money. I'm sure his agent touched base with a number of teams that KT would consider playing for.