Every connected compact Hausdorff space of weight $\aleph_1$ is the remainder $v \mathbb R \setminus \mathbb R$ of some compactification of $\mathbb R$. In particular, $[0,1]^{\aleph_1}$ is the remainder of a compactification of $\mathbb R$, and therefore $\mathbb R$ has a compactification with remainder of cardinality $2^{\aleph_1}$.
Using forcing, it is possible to show that it is consistent to have $\mathfrak{c} < 2^{\aleph_1} < 2^{\mathfrak{c}}$. (For example, Easton's Theorem immediately implies that we may get a model where $2^{\aleph_0} = \aleph_2$, $2^{\aleph_1} = \aleph_3$, and $2^{\aleph_2} = \aleph_4$, although Easton's Theorem is a bit overkill for this.) Thus it is consistent that $\mathbb R$ has a compactification with cardinality in $[\mathfrak{c}^+,2^{\mathfrak{c}})$.
(The result about weight-$\aleph_1$ continua is proved by Dow and Hart in this paper. But the special case of $[0,1]^{\aleph_1}$ is actually much easier to prove, using the fact that $[0,1]^{\aleph_1}$ is separable. Let $\{d_1,d_2,d_3,\dots\}$ be a countable dense subset of $[0,1]^{\aleph_1}$. Map $\mathbb R$ into $[0,1] \times [0,1]^{\aleph_1}$ as follows. First map $\mathbb R$ onto the ray $[1,\infty)$, and then map $[1,\infty)$ into $[0,1] \times [0,1]^{\aleph_1}$ by linearly mapping each interval $[n,n+1]$ to the line segment connecting $(d_n,\frac{1}{n})$ to $(d_{n+1},\frac{1}{n+1})$ in $[0,1] \times [0,1]^{\aleph_1}$. This mapping embeds the ray $[1,\infty)$ in $[0,1] \times [0,1]^{\aleph_1}$, and its boundary in this embedding is precisely the set $\{0\} \times [0,1]^{\aleph_1} \approx [0,1]^{\aleph_1}$.)