Improvements ...

How improve convection heat transfer ?

Two angles of attack arise naturally : one relates to h and the other to S, even both simultaneously. It is a question of increasing both in order to decrease the convection thermal resistance.

Heat-transferring surface area : fin

The "Simplest" method is to increase the area of the plate with a bigger one, but this is limited by the fact that the spreading resistance also increases, with its related drawbacks.
The best method to increase contact surface area between fluid and solid is to use fins, and the vast majority of coolers are provided with such. Rather than adding surface area while spreading out horizontally, it is done vertically.

On the practical and theoretical side, the effectiveness of a fin will primarily depend on its thickness and height. Each part of the fin surface will not exchange the same amount of heat, since the temperature decreases from the base to the top of the fin.
This results in a coefficient (efficiency, ranging between 0 and 1) which will balance the fin-fluid contact surface area, so that the fin's heat transfer with the fluid occurs on an effective area smaller than the real contact area.
Generally, beyond the point where the temperature of the fin is nearly identical to that of the fluid, the remainder of surface exchanges virtually no heat to the fluid and is thus useless. Moreover, the higher fins are, the more the volume of the cooler is and if the flow rate is constant then its velocity decreases, also the thermal convection ... it's a matter of compromises, as in most situations.

Other improvements:

They relate primarily to h but can also have a slight positive impact on the heat-transferring surface area. They generally consist of mechanisms to "break" the boundary layer in order to reduce it, to disturb the flow so that mixing is increased as well as turbulence.
In practice this is done by tweaking the geometry and the finish quality of the heat-transferring surfaces. Rough surfaces and the presence of turbulators will achieve this task.

But always keep in mind that the benefit in performance obtained thanks to these improvements will be paid on the other hand by pressure losses which will be addressed further ... compromises ...