It is often easier to construct *Galois* covers of degree $n$. 

In the case where $E$ is an elliptic curve and we have $t$ branch points, by Riemann existence theorem a Galois cover with Galois group $G$ is defined by the following data: 

$\bullet$ a finite group $G$ of order $n$;

$\bullet$ elements $a, b, g_1, \ldots, g_t$ such that $G= \langle a, b, g_1, \ldots, g_t \rangle$ and $[a,b]g_1g_2 \ldots g_t=e$.

An important remark is that for a Galois cover the local monodromies around points in the same fibre are *conjugate* transpositions, i.e. points over the same branch points have monodromies with the same cyclic structure (this is in general false for non-Galois covers). Moreover, over the branch point $b_i$ such a structure is a cycle of order equal to the order of the element $g_i$.

Let us see now how to construct Galois coverings in the cases you are interested in. 

**Three branch points, that is $t=3$**. For each $n$ we can construct a cyclic cover of degree $n$: simply choose $$G=\mathbb{Z}/n \mathbb{Z}=\langle x | x^n=1 \rangle $$ (I use the multiplicative notation) and take $$a=x, \quad b=x, \quad  g_1=x, \quad g_2=x, \quad g_3=x^{-2}.$$

**Two branch points, that is $t=2.$** Again, we have a cyclic cover for any $n$. Choose $G$ as above and take $$a=x, \quad b=x, \quad  g_1=x, \quad g_2=x^{-1}$$.

**One branch point, that is $t=1.$** A moment of reflection shows that we cannot take $G$ abelian in this case, otherwise $[a,b]g_1=e$ implies $g_1=e$, that is the cover would be unramified, contradiction. We can however construct a *dihedral* cover of degree $n=2m$
 for any $m$. In fact, set $$D_{2m}= \langle x, y | x^2=y^m=1, \quad xy=y^{-1}x \rangle$$ 
and take $$a=x, \quad b=y, \quad g_1=y^2.$$ 

regarding your last comment, observe that when $m=5$ you have a Galois cover of degree $10$ with a unique branch point, whose branching order is $5$. In other words, over the unique branch point $b$ one has precisely two points $q_1, q_2$ and the local monodromy around each of the $q_i$ is by construction a $5$-cycle. 

One can also construct Galois covers of odd degree with a unique branch point, for instance taking more general metacyclic groups (i.e. non-trivial semidirect products  of cyclic groups). The dihedral covers are in fact a particular case of this construction.