Is eigenfunction for angular momentum operator is unique?

I want to know if the commutation relation for angular momentum, $$left[ J_alpha,J_betaright] = ihbarepsilon_{alphabetagamma}J_gamma$$
is enough for defining its unique eigenstate of spin, $lvert s,mrangle$.

As far as I know, the relation is enough for defining ladder operator $J_pm$.
In the case of orbital angular momentum L, ode $L_+psi_t=0$ and $L_-psi_b=0$ defines unique state, thus uniquely defining the whole ladder $lvert l,mrangle$.

However, in the case of spin, the state vector lies in abstract space, so the equation $S_+psi_t=0$ and $S_-psi_b=0$ can’t guarantee the uniqueness of $psi_t$ and $psi_b$. Moreover, the relation $S_z(S_+lvert s,mrangle)=(m+hbar)(S_+lvert s,mrangle)$ shows that $S_+lvert s,mrangle$ is one of eigenstate for $S_z$ with eigenvalue $m+hbar$, but this may not be unique.
Isn’t there a possibility that $S^2$ and $S_z$ have some degeneracy? If then, the quantum ladder may not cover the whole basis for $S^2$ and $S_z$.

I think ignoring these possibilities, and regarding these procedure as unique construction of $lvert j,m_jrangle$ will be enough, but I want a strict demonstration. Does this have something to do with generator-nature of angular momentum?

Edit) Considering other index k, such as position $vecr$, will give a state vector $psileft( r right) lvert s,mrangle$ which will generate eigenvectors with same eigenvalue as well as different spacial component $psileft( r right)$ is given.

What I’m wondering precisely is :
The standard theory for angular momentum, starting from the commutation relation, will in the end argue the existence of eigenstate $lvert j,m_jrangle$, satisfying
$$J^2lvert j,m_jrangle = jleft( j+1 right)hbarlvert j,m_jrangleJ_zlvert j,m_jrangle = m_jhbarlvert j,m_jrangle$$
I’m wondering if it is possible to find $lvert j,m_jrangle’$ without introducing any other index.
In the case of angular momentum L, it is possible to directly solve the eigenvalue equation to find the unique solution(spherical harmonics), so the problem I’m wondering happens on the case of spin S, where algebraic relation is the only restriction.