Increasing partition coefficient in an extraction
Generally, extraction of substances with organic solvents is influenced by various factors, among which there are the following:

nature of the extracted substance and the extractant (not a variable in your example);
temperature;
presence of electrolytes in aqueous solutions;
$\mathrm{pH}$ of the medium;
rate of agitation etc.

In your particular case probably only temperature and electrolytic presence are relevant.
Change in temperature often affects the distribution constant rather seriously, as not only the solubility of the extracted substances in each phase changes (unequally), but also the mutual solubility of the organic and aqueous phases changes too. When the temperature changes, the dissociation/association of the substance in the corresponding phase can change, influencing hydration (solvation) and therefore extractability. Ideally you might want to check solubility-temperature plots for your system, taking into account that this data often cannot be obtained by extrapolation.
Addition of highly soluble salts (electrolytes) to an aqueous solution of another substance can lower (salting-out effect) or increase (salting-in effect) its solubility in water. Increased high ionic strength also lowers activity of water in terms of depressing its competitive effects (first of all, solvation).
The salting-out effect of electrolytes depends on the nature and properties of the extracting substance and the concentration, charge and ionic radii of the salting-out ions. Smaller ions with larger charge have higher charge density than ions with a larger radius, therefore they are hydrated better.  $\ce{NaCl}$ and $\ce{Na2SO4}$ are commonly used for salting-out organic substances. However, this rule has a number of exceptions. Some non-electrolytes that are well soluble in water also have salting out-effect. For example, ethyl alcohol salts-out acetic acid from its aqueous solutions upon extraction with ethyl acetate.
The salting-in/out phenomenon is often explained based on the electrostatic model proposed by Debye and McAulay, reviewed in [1], and can be approximately described via equation:
$$c/c^\circ = 1 - K_D \cdot c_\mathrm{s} + A \cdot c_\mathrm{s}^{0.75},$$
where $c^\circ$ and $c$ are solubilities [$\pu{mol L-1}$] of non-electrolyte in pure solvent and solvent with salt, respectively; $K_D$ and $A$ - empirical coefficients; $c_\mathrm{s}$ - salt concentration.
Pretty much your explanation is on the same page with the original Debye theory:

This theory pictures the &quot;salting-out&quot; effect as a consequence of the aggregation of the water molecules of greater polarity around the ions owing to the field of the latter. The less polar non-electrolyte molecules tend to congregate in the portions of the solution most remote from the field of the ions, with the net result that an enhancement of the water molecules around the ions and of the non-electrolyte in the solution regions away from the ions occurs. A reduction of the solubility of the non-electrolyte, referred to the total water present, thus results, due to the increase of non-electrolyte-water ratio in those regions of the solution containing the non-electrolyte.
The criterion of polarity which Debye employs is the dielectric constant of the saturated aqueous solution of the non-electrolyte relative to water at the same temperature. On this basis if the saturated solution has a dielectric constant less than water, &quot;salting out&quot; occurs in the presence of the added salt. If the dielectric constant of the saturated non-electrolyte solution is above that of water, the non-electrolyte becomes more soluble in the salt solution or is &quot;salted in&quot;. This means that the non-electrolyte tends to aggregate around the ions at the expense of the water, and so the nonelectrolyte-water ratio in regions removed from the ions is lowered and the total water present can hold more non-electrolyte in solution.

Also, in biochemistry so-called Hofmeister series are used to favor protein extraction by salting-out for given conditions.
References

Gross, P. M. The “Salting out” of Non-Electrolytes from Aqueous Solutions. Chemical Reviews 1933, 13 (1), 91–101. https://doi.org/10.1021/cr60044a007.