32 states had removed the stay-at home orders by 2 weeks ago. 21 of them have seen cases remain steady or fall. 10 states had removed them by 30th April, including Texas. 5 of the early openers - Tennessee, Montana, Idaho, Georgia and Colorado - have seen cases remain steady or fall for well over a month. 2 more have seen cases fall continuously in June. A blanket comparison based on the stay-at-home orders is limited as some restrictions remained after them, with little consistency across states, so it's easier to look at particular cases.
Iowa relaxed restrictions in 77 of the state's 99 counties on the 1st May, and all of them on the 15th. Cases fell from a 7-day average of 500 (cases) on the 1st May, to 384 on the 15th May, to 319 yesterday.
In Kansas, all businesses besides bars, clubs and museum-types were re-opened on the 4th May, and by the 15th May they allowed museum-types to open. On the 4th May they have a 7-day moving average of 266, by the 15th it had fell to 178, and yesterday it was at 103.
In North Dakota, all businesses besides sports and music events were re-opened on the 1st May. The 7-day moving average was at 52 then, now it's at 42.
It just isn't the case that two weeks after restrictions have been relaxed, there's been a subsequent increase in cases. This is the complete overview for states that have removed the stay-at-home orders. Some go up, some go down, some stay stable.