The phrase "almost instant access" is correct and usable in written English.
You can use it to describe a situation where access to something is very quick, but not entirely immediate.
Example: "With the new app, users can enjoy almost instant access to their favorite content."
Alternatives: "nearly immediate access" or "rapid access".
Lenders got almost instant access to their cash at a higher yield.
Never before has it been so easy to start a business — and have almost instant access to more than a billion people for whatever you create.
It's also possible to chain skill kills together, which not only gives almost instant access to abilities, it also multiples your score count for the level.
Threadneedle Street cited these open-ended commercial property funds, which offer almost instant access to cash, as having the potential to exacerbate problems in the market.
The AudioReQuest packs a 20-gigabyte hard drive, which can give consumers almost instant access to more than 6,000 stored songs.
Mr. Sharpton, who arrived late but was given almost instant access to the microphone, said that livery cabs were a means to move about in areas where yellow cabs did not venture.
To get out ahead of nature, the WOF will, to begin with, offer instant access to almost every environmental variable that science can measure, from points scattered all across the face of the planet.
Access was almost instant, with no waiting lists and services available to all people living at home.
We have instant access to an almost infinite library of work by other artists, researchers, developers, and it has become one of the most efficient learning tools, allowing anyone online to understand the different steps or parts of a creative process.
Owing to electronic publishing, scientists today have instant access to an almost limitless amount of research articles (Björk et al., 2009).
The best five-year fix, from Paragon Bank, pays 3.06% – almost double the best rate on an instant access account.
