Would it be recommended to insert 5 or 6 letters of recommendation into my Army OCS packet?

I threw in one extra for my WOCS application, worked out well (from a MG). I believe when it comes to extra LORs that it is more of the quality and "who" wrote them, rather than how many you have. 3 extra LORs from teachers, CPTs, etc... they better write really well, an extra from a General/CW5/Congressperson, etc.. that'd make a statement.

As long as you have at minimum 3 letters of recommendation you will be fine. What makes you competitive isn’t so much the letters but everything else. The APFT, your ASVAB, college are some but the biggest one is just you. When you are sitting in front of the board be confident because how you portray yourself will carry more weight than anything in your packet.

Quantity is nothing. Quality is everything.