No matter how good your impulse engine is, you cannot go FTL without activating warp engines.
That would depend on how the impulse engine works.
Is it just a sophisticated fusion rocket engine? Or is it a "baby" warp drive of some sort, optimized for slower than light travel, but capable of minor FTL in a pinch?
The latter would make the Enterprise's ability to reach Delta Vega is WNMHGB easier to understand.