No. Don't listen to anyone recommending using PEX. Worst product on the market in 50 years. I have 30-50 cases a year associated with determining why PEX fittings, PEX lines, PEX crimps have failed and flooded the house. There are multiple class action lawsuits against the various manufacturers of PEX plumbing fixtures.
If you have pinhole leaks in your copper, it can be a number of different issues, depending on whether it's the hot water lines or the cold water lines. Also depends on whether you have soft water or hard water. There is a treatment, depending on what you have going on in your house, that will flush the system and then coat the interior of the pipes to seal against future pinhole leaks. This is a controversial system and I haven't heard it used in 10 years. If at all in doubt, replumb.
Depending on what is causing the pinhole leaks, your vertical lines may be as at risk as your horizontal runs.