*sighs*
Rooks, Suppressive Fire specifically says it lasts until the end of the Combat Turn. You're thinking of Initiative Passes within a Combat Turn; and yes, you could theoretically do that, but you would move the area each time, not overlap.
Having 4 Initiative Passes is not exactly trivial by the way. To get 4 passes you need an Initiative Score of 31 or more. With a maximum initiative dice of 5 you can at most get 30 if you roll all sixes and you're more likely to be rolling in the 10 to 25 range. To reliably get 31 or more you'd need a combined Reaction and Intuition score of 13 or 14, which means hardcapping Intuition at 6 and getting some Reaction boosters to get Reaction up to 7 or 8. Getting that 5th Initiative Dice isn't trivial either...