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In a conductor, all the electrons are few to move to conduction bands.

If we say that magnitude Current is increased (I= dq/dt) can we infer that the velocity with which the electrons flow also increases if all the electrons are already involving in conduction?

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Considering by velocity of electrons you mean the average velocity, aka Drift Velocity, You are correct.

Infer this physically, like with more current then the charge per unit time must increase, and the wire is already saturated with charge so only option is for their speed to increase, Or see it mathematically,$$j=nev_d$$ Increase $j$($ \Rightarrow$ increase current) would increase $v_d$.

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