Can we increase speed of electron in order to increase electric current in a particular substance(assuming it allows electricity to pass)?

drift velocity Vd
Vd = I / nqA
Vd is drift velocity in m/s
I is current
n is the number of charge carriers per m³
for copper, 8.5e28 electrons per m³.
A = πr² is the cross sectional area in m²
q is the charge of the charge carriers (electrons)
q = –1.602e-19 Coulomb (charge on an electron)

drift velocity is the actual speed of the electrons as they travel down the wire. It is very small, but it is proportional to current, as you see. Can be down in the inches per second or even slower.

When you turn on a battery, the current flow appears very fast, almost at the speed of light. But that is due to the NET electron flow. Visualize it as a pipe filled with ping-pong balls. push a new one into one end and one pops out the other end quickly. But the actual balls move slowly down the pipe.

Absolutely. By definition 1 ampere == 1 Coulomb/sec; so in I = V/R when we increase the voltage V over a resistance R (e.g., the wire) we get a higher current, which mean more C/sec flowing. And there are roughly 1E19 electrons in a Coulomb; so you are actually pushing through more electrons when the current rises.

Understand the electron density is not increased. It's the speed of the electrons there that increases to make the higher current flow.

Yes. By increasing the applied voltage we are increasing the electric field. This increases the electron speed.