Relative speed when getting close to the speed of light

There are at least two misunderstandings in your argument. The most fundamental one is that you tried to compare times in two different reference frames, but you can't do that in relativity. For example, if Bob zooms off at $frac{1}{2}c$ to the right, and Alice stays where she is, then both of them will see "time dilation" when they look at the other person. Suppose that they are both carrying clocks. Both of them will think that the other person's clock is running slower than their own.

This comes into your scenario because I think you are conflating time dilation observed from the particle moving at $0.01% c$ and time dilation from the rest frame.

The second misunderstanding here is that observers will not only see time dilation, but also space dilation, so this also makes figuring out the relative velocities a little bit harder.

Doing this calculation properly leads to the velocity-addition formula. Suppose that you have $v_1=0.9999c$ and $v_2=0.0001c$. Then the velocities that these two particles will perceive each other to be moving at (if they could perceive things and take measurements!) would be $$v_text{rel} = frac{v_1+v_2}{1+frac{v_1 v_2}{c^2}} = frac{c}{1+0.9999 cdot 0.0001} approx 0.99990002 c $$