In Doppler cooling procedure, is it theoretically possible to use the emitted photons from the target atom (cloud) to cool more atoms? The idea is to cool the atoms rapidly by using energy emitted by other atoms in a multiplier effect of sorts.
In the event of spontaneous emission (the one responsible for Doppler cooling) a photon is emitted in a random direction. If you have 100 % collection efficiency and combine these "recycled" photons with the original beam, then it should be possible. Theoretically it is even possible to correct the polarization and wavelength of such photons depending on the region of space where they are collected.
The short answer is no.
The magic that makes Doppler cooling work is that the atoms only absorb photons moving exactly opposite the direction of the atom's motion, thanks to the Doppler shift. You need the precise relationship between the photon momentum and the atom momentum for the absorption to happen such that the net effect is to slow the atom.
The spontaneously emitted photons have a random direction and also a slightly higher energy than the absorbed photons. If one of these photons is absorbed by another atom, it will cause heating because the precise conditions for cooling aren't satisfied. This rescattering effect is one of the practical limits to the minimum temperature and maximum density of an optical molasses or magneto-optical trap. (The other limit being the optical density of the atom cloud.)
You can also think of this from a thermodynamics point of view. Cooling requires removing entropy from the atom cloud. This entropy is carried away by the spontaneously emitted photons as randomness in their direction and energy. If you redirected them back to the atoms they would act like a hot (relatively speaking) temperature bath.
This entropy in the spontaneously emitted photons also makes it impossible (without doing work) to correct for the polarization, energy, and direction, as @Rol suggests.