Fence hedge compatability

We are building an extension but in order to do this a section of the neighbours boundary hedge will need to come out. For years the owners said they hated it and would rather a fence . We are offering to replace the removed section with a fence of their choosing, but they seem to have developed a sudden rush of affection for a specific section of the hedge. It surrounds their decking at the back and the concern is of privacy as well as affection. I have offered to replace that section with fence as soon as the builders have finished but then it was about a foot gap between the decking and a new fence. I know this may be an unlikely pairing but is there a  hedge that would sit well with a fence up against it that would grow quickly. The neighbours are as a rule great and have offered part of their garden as parking whilst the build is ongoing so I'm desperate to please.