Hembery emerged from Sunday's German Grand Prix breathing a sigh of relief after the weekend passed smoothly compared to what transpired at Silverstone seven days previously when a series of high-speed blowouts placed the lives of drivers, marshals and fans in danger.
Pirelli introduced a new-specification of rear tyre that included a belt made of Kevlar - a high-strength synthetic fibre which is more resistant to punctures - as opposed to the steel previously used.
Later this month, from the Hungarian GP onwards, Pirelli will supply a new range of tyres that comprise last year's structure with this year's compounds, which are to be tested at Silverstone next week during the young driver test.
Beyond that, Hembery concedes Pirelli is up against the clock if they are to supply tyres for next season when new engine regulations come into force.
"It was obviously very important for us to have a clean weekend in Germany, which goes without saying," said Hembery, speaking to Press Association Sport.
"We had to react quickly, we're serious about what we do. We don't want days like Silverstone, obviously.
"It was important we showed we can react and do something that's right, and we'll continue to do that.
"But we still want a number of changes made to the way we're working in the sport, the way we're able to test.
"We've two (private) test sessions coming up and we're running around in a four-year-old car which is six seconds slower (compared to this year's models).
"That's clearly far, far, far from acceptable.
"There are also still contractual things up in the air. There are still many, many things that need to be done."
A week ago Pirelli were informed by the FIA and F1 supremo Bernie Ecclestone they could use a current car to test.
Despite Pirelli having total control of any test, and a team not being allowed to run any new parts, paranoia is high in F1.
If one team helped Pirelli, their rivals would deem they are gaining an advantage, so leaving the company in limbo.
Pirelli were also due to run a test for two days at Interlagos following the season-ending Brazilian GP, but at the end of a long campaign, again no team is willing to help.
"Ahead of the opening race of next year there's still so much work to be done, and we could have huge surprises with the new powertrains," added Hembery.
"If we don't get the testing sorted out...and I was surprised no-one even thought about going wet testing before we get to the new season.
"With the torque the new engine is going to have, in going somewhere like Malaysia and going into a wet session for the first time with a new car, I don't think the drivers are going to be very pleased if the range of wet-weather tyres have not been tested properly.
"There is an awful lot of detail that needs to be done, and we need to find a way of getting solutions the teams can live with, but allow us to deliver what we need to.
"Unfortunately, there is still a long way to go."