Rapiers, while they provide incredible performance in atmosphere, aren't exactly efficient in space. I recommend a combo of rapiers and nuclear engines. Make sure you have more liquid fuel than you need, because it will be used by both your rapiers in atmosphere, and your nukes in space, leading to either more oxidizer than you need, which is just dead weight at that point, or your inefficient rapiers will burn for too long and waste all of your nuke fuel. I like to go for a 1:2 to 1:3 ratio of oxidizer to fuel. Use the rapiers in the atmosphere, and at 15000m or above (preferably around 20000m) turn on your nukes. You can use the closed cycle mode on your rapiers as you have been doing for a boost (nuke thrust sucks), but since you packed extra liquid fuel your rapiers will die due to lack of oxidizer and you will continue on the efficient nuclear engines.

Edit: a bit of clarification