The Brackley-based squad enjoyed a much more fruitful Canadian Grand Prix compared to other races this campaign, with George Russell securing pole position before finishing in third, one place ahead of team-mate Lewis Hamilton.
Mercedes' cause was assisted in Montreal by the recently repaved track surface, which provided a smooth platform for the team to dial the car into – a common theme at various track layouts in the past two-and-a-half seasons.
Whilst steps have been taken to widen the operating potential of its machinery, Allison teased a raft of new parts coming in the future to 'bully' the car into shape.
"The changes we have made are making this car a better car and that will be true at every circuit we go to," he explained.
"The characteristics of Montreal make it look a little quicker than we have a natural right to command at the coming races. In Barcelona, I think it more likely that we will be competitive, but not right at the front because the next tracks are a little bit of a sterner test of a car. Hot asphalt, wider cornering speeds.
"However, I also know what we have got coming. I also know what we are planning to further improve the car.
"Our challenge is just to keep those upgrades arriving at a pace that the others cannot keep up with and in doing that, just bullying our car to the front by virtue of the effort made by everybody here over the coming weeks and months to get the car so that it can have its Montreal weekend or better at any track that we face in the future."
One of the key issues with each of the three cars developed by Mercedes so far under F1's ground effect regulations has been that they have only had a narrow operating window to achieve optimum performance. But this is something Allison believes is starting to be addressed.
"I think we have broadened it [the window] substantially," he insisted.
"There is more we still need to do, and we will know for sure when we go to the next track, which is Barcelona, because there is really a very substantial range of cornering conditions.
"There is also a much hotter track, and so that will be quite a stern test of a vehicle."
A major component in the upturn in form was a new front wing added to the W15, which is a more conventional design compared to the initial iteration run this season and on whether it worked as expected, Allison said: "I would say yes.
"We had got an idea of how it would behave because we had run it the previous race in Monaco with just George on that occasion. We had two of them in Montreal, and we expected it to perform well.
"We expected it to deliver a bit more in Canada than it did in Monaco, because the Canada circuit, although unusual, is more of a normal circuit than Monaco was.
"It delivered more performance, it made the car feel easy to drive, well-balanced, and made the car the driver's friend rather than the thing they had been fighting, which has been what has been problematic in the opening part of the season for us."
