While there is cause for optimism, unemployment remains the biggest concern for the sector, particularly when the government's furlough scheme comes to an end.
Other, smaller parts of the economy, also offered some hope - in particular manufacturing.
Manufacturing PMI was recorded at 50.1 in June, above 40.7 in May.
IHS said: "June data indicated a vastly improved overall picture across the UK private sector, with the downturn in total business activity continuing to steady after the record rate of decline seen at the height of the lockdown during April."
Duncan Brock, Group Director at CIPS, said: "At the end of the second quarter we saw some uplift in the UK private sector as lockdown eased, businesses started opening up and activity increased from the crippling lows of the last few months.
"Though the index still remained below the no-change mark in June, its rise since May was a survey record, creating another signal that some bounceback has started to emerge following April’s historic lows.
"Manufacturing was the strongest of the two registering above the no change mark and optimism rose to its highest levels since September 2018."
