Gravitational potential energy sign

Following is a small derivation just so I can explain my question. The gravitational potential energy is:

$$(*)U_g = -frac{GMm}{r}$$

And:

$$ Delta U =-GMm(frac{1}{r_{final}} – frac{1}{r_{initial}}) $$

If some mass $m$ is taken a height $h$ above the ground, we get:

$$ Delta U =-GMm(frac{1}{R+h} – frac{1}{R}) = frac{GMmh}{R(R+h)} $$ approximating $hll R$ :

$$ Delta U = frac{GMmh}{R^2} $$ and if we denote $g=frac{GM}{R^2}$ we get the familiar $$ Delta U = mgh$$

That indeed goes hand-in-hand with (*), since the object went further from the center of the earth and therefore gained PE.

Now to the question: Does that mean we should always express the PE to be "more negative" the closer we are to Earth? I see some texts that present PE that gets bigger when you get closer to the Earth and that quite confuses me.