The only time I saw slack in the new chain was when the tensioner was out, or backed way off OR when moving the crank CCW the slightest amount.
I am unclear if you are asking about installing a new tensioner, or making comment about a new tensioner that is installed. By all means install a new tensioner with a new chain AND a new tensioner will not be providing tension (beyond the spring tension) until the engine is running and oil pressure is up.
'73 280SEL 4.5 (9/72)- RIP
Only 8,173 units built from 5/71 thru 11/72
'02 CLK320 Cabriolet - wifey's mid-life crisis
2012 VW Jetta Sportwagon TDI...at least its a diesel
Non illegitemae carborundum.