  This study reconsidered the N=1 supersymmetric extension of the W_3 algebra
which was studied previously. This extension consists of seven higher spin
supercurrents (fourteen higher spin currents in the components) as well as the
N=1 stress energy tensor of spins (3/2, 2). Thus far, the complete expressions
for the higher spin currents have not been derived.
  This paper constructs them explicitly in both the c=4 eight free fermion
model and the supersymmetric coset model based on (A_2^{(1)} \oplus A_2^{(1)},
A_2^{(1)})$ at level (3,k). By acting with the above spin-3/2 current on the
higher spin-3 Casimir current, its fermionic partner, the higher spin-5/2
current, can be generated and combined as a first higher spin supercurrent with
spins (5/2, 3). By calculating the operator product expansions (OPE) between
the higher spin supercurrent and itself, the next two higher spin supercurrents
can be generated with spins (7/2, 4) and (4, 9/2). Moreover, the other two
higher spin supercurrents with spins (4, 9/2) and (9/2, 5) can be generated by
calculating the OPE between the first higher spin supercurrent with spins (5/2,
3) and the second higher spin supercurrent with spins (7/2, 4). Finally, the
higher spin supercurrents, (11/2, 6) and (6, 13/2), can be extracted from the
right hand side of the OPE between the higher spin supercurrents, (5/2, 3) and
(4, 9/2).
