It's not clear to me what you are trying to do. Are you...
1. trying to increase print contrast overall in both highlights and shadows, but when you do, the highlights get lighter? If so, just increase exposure to bring the highlight values to where you want them. If the shadows are then right, you've got the right contrast. If not, then change contrast (and exposure) again. As suggested, if you change contrast, a new test strip is usually required to find the right exposure time for the highlights.
2. trying to keep the contrast you already have in the upper midtones and highlights while increasing the contrast of the lower mid-tones and shadows? This is a more difficult situation. Here's where split-grade printing techniques will help. If you have the print exposure correct and the highlights/upper mid-tones where you want them but need to boost contrast in the shadows, then I'd burn the shadows with a higher-contrast filter. This will deepen the blacks without affecting the mid-tones and lighter areas in those areas much.
Or, you can give the print more exposure with a higher-contrast filter, letting the highlights go too white, and then burn the highlights back down with a lower contrast filter. This is much the same as above, but can be subtly different. The choice is yours.
FWIW, I find that with most VC papers, the contrast in the extreme highlights, i.e., the areas just almost white but not quite that are down low on the toe of the paper, doesn't change much in contrast compared to the mid-tone and lower values when you change contrast filtration. This is something that I miss very much about premium graded papers of the past. It's hard to get subtle separations in the highlights with VC paper while keeping those values close to paper-base white. I find I have to print down a bit more than I'd really like to. The old Oriental Seagull G could get the whitest parts paper-base white and then give a quick contrast change to the next lower value. I use bleach more now.
Best,
Doremus