Try this out - I don't have your custom objects/fields so this hasn't been compiled or tested, but it looks better.

1. -- Use a list of PI records so that you only insert them all once
2. -- Use a list of Contact records that need to be updated after the PI records have been created
3. -- This was a bit tricky because we have to insert the PI records before we know what their ID is, and then want to retrieve the id corresponding to the Contact list we cycle through.  I used a Map to link a Contact id with its matching PI id.

You might need to play around to get it to compile but hopefully this will get you on the right path.

trigger insertPIOPrec on Contact (after insert) {
// make a list of Contacts that need an additional update
List<Contact> ContactList = new List<Contact>();
// make a list of Contacts with updated data
List<Contact> ContactUpd = new List<Contact>();
// list of PI records to create
List<PI_Ongoing_Participation__c> NewPI = new List<PI_Ongoing_Participation__c>();

// cycle through inserted Contacts to find which need a PIO created
for(Contact c: trigger.new) {
If (c.PI_Ongoing_Participation__c == null){
PI_Ongoing_Participation__c p = new PI_Ongoing_Participation__c(Contact__c = c.Id);
p.Contact__c = c.Id;
NewPI.add(p);
ContactList.add(c);
} //end if
} // end for loop
// insert PI records now
insert NewPI;

// make a map of Contact id to new PIO id records to use for populating Contact
Map<Id,Id> PIMap = new Map<Id,Id>();
// cycle through new PI records and build out map
for (PI_Ongoing_Participation__c pi : NewPI) {
PIMap.put(pi.contact__c, pi.id);
}

// finally go through Contact list and populate the PI field
for (Contact updC : ContactList) {
updC.PI_Ongoing_Participation__c = PIMap.get(updC.id);
ContactUpd.add(updC);
} // end for
// now update the Contacts with the new PI record
update ContactUpd;
}