Plotting particle movement Time/keyframe issue

I have data with format as below:

 x               y             z            time T   
 -71.7087326    40.2462502    5.02939129    2.31992912
 -67.7142334    40.249218    5.09217882    5.15146351
 -61.1967964    40.2587852    5.23078823    9.7579298

I want to move a particle so that at time T, it will be at position x, y, z accordingly.

Currently what I do is to calculate average speed between 2 points (distance/time) and move the particle accordingly. Below is my code:

 class Data
 {
     Vector3 pos;
     float exactTime;
 }
 
 class CurrPoint
 {
     int id;
     float currTime;
     Vector3 avgSpeed;
 }
 
 CurrPoint p;
 Data data;
 
 void Update()
 {
     if(p.currTime == 0)
     {
         point.position = data[p.currID].pos;
     }
     
     p.currTime += Time.deltaTime;
     
     if (p.currTime < data[p.id].exactTime) 
     {
         point.position += Time.deltaTime * p.avgSpeed[p.id];
     } 
     else 
     {
         
         //exceeding value
         //not the last node
         if (p.id < data.Count - 1) 
         {
         
             //calculate time needed to complete to next pos
             float deltaT1 = data[p.id].exactTime - (p.currTime - Time.deltaTime);
             float deltaT2 = Time.deltaTime - deltaT1;
                 
                 
             //move to next id, calculate distance
             p.id++;
             points [i].position = data[p.id].pos;
             
             //move using new speed as much as deltaT2
             points [i].position += deltaT2 * p.avgSpeed [p.id];
         } 
         else 
         { 
             //last node
             //go back to beginning again (Reset)
             p.id = 0;
             p.currTime = 0;
         }
     }
 }

And here is what it is like when the game run: [http://youtu.be/1llmgJHNtPw]

As you can see, there are times when the particle will be 'pushed back' before moving again. I think it is because the time/framerate that is not right as I am using Time.deltaTime. The particle has travelled too far before processing data at next line.

How to ensure that this won't happen? Or perhaps if there is other suggestion, you are welcomed. Thanks a lot!