We are concerned with a nonlocal transport 1D-model with supercritical dissipation
γ
∈
(
0
,
1
)
\gamma \in (0,1)
in which the velocity is coupled via the Hilbert transform, namely the so-called CCF model. This model arises as a lower dimensional model for the well-known 2D dissipative quasi-geostrophic equation and in connection with vortex-sheet problems. It is known that its solutions can blow up in finite time when
γ
∈
(
0
,
1
/
2
)
\gamma \in (0,1/2)
. On the other hand, as stated by Kiselev (2010), in the supercritical subrange
γ
∈
[
1
/
2
,
1
)
\gamma \in \lbrack 1/2,1)
it is an open problem to know whether its solutions are globally regular. We show global existence of nonnegative
H
3
/
2
H^{3/2}
-strong solutions in a supercritical subrange (close to 1) that depends on the initial data norm. Then, for each arbitrary smooth nonnegative initial data, the model has a unique global smooth solution provided that
γ
∈
[
γ
1
,
1
)
\gamma \in \lbrack \gamma _{1},1)
where
γ
1
\gamma _{1}
depends on the
H
3
/
2
H^{3/2}
-initial data norm. Our approach is inspired by that of Coti Zelati and Vicol (IUMJ, 2016).