As Mohammad Golshani remarked, it is possible to control the cofinality of $j(\kappa)$ by iterating the forcing that adds a function $f\colon \kappa \to \kappa$ which is eventually larger than any ground model function.

The conditions of the forcing notion are pairs of the form $(s, g)$ where $s{\in} ^{<\kappa}\kappa$ and $g\colon \kappa \to \kappa$, where $(s, g)$ is stronger than $(t, h)$ if $s\supseteq t$, $g \geq h$ everywhere and $\forall \alpha \in \text{dom } s \setminus \text{dom }t$, $s(\alpha) \geq h(\alpha)$.

Assuming $\kappa^{<\kappa} = \kappa$, this forcing is $\kappa$-centred and $\kappa$-directed closed. Let $\mathbb{P}_\alpha$ be the iteration of adding dominating function for $\alpha$ many steps with support ${<}\kappa$. Using the $\kappa$-closure of iteration and standard $\Delta$-system arguments - this iteration is $\kappa^{+}$-c.c. Therefore, it doesn't collapse cardinals.

Let $\langle f_i \mid i < \alpha\rangle$ be the sequence of the generic dominating functions.

**Lemma:** If $\text{cf }\alpha \geq \kappa^{+}$ then the true cofinality of $\kappa^{\kappa} / J^{bd}$ is $\text{cf }\alpha$. Namely, there is a cofinal, increasing sequence of functions in $\kappa^{\kappa}$ of order type $\text{cf }\alpha$.

**Proof:** Let $\lambda = \text{cf }\alpha$. Let $\langle \gamma_i \mid i < \lambda\rangle$ be a cofinal sequence. The sequence of functions $g_i = f_{\gamma_i}$ is increasing (modulo bounded error) and by the chain condition of $\mathbb{P}_\alpha$, every function in the generic extension is bounded by one of them. $\square$

Let $\kappa$ be a measurable cardinal. If $\text{tcf } \kappa^{\kappa} / J^{bd} = \lambda$, then for every $\kappa$-complete measure $\mathcal{U}$ on $\kappa$, $\text{cf }j_{\mathcal{U}}(\kappa) = \lambda$. Therefore, in order to construct a model in which $\text{cf }j(\kappa) = \lambda$, $2^\kappa = \mu$ where $\kappa < \lambda \leq \mu$, $\lambda$ regular, $\text{cf } \mu \geq \kappa^{+}$, we may start with indestructible supercompact $\kappa$ such that $2^\kappa = \kappa^{+}$ and force with $\mathbb{P}_{\mu + \lambda}$.

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