Do indiscernibility embeddings exist for an initial segment of an inner model of many measurable cardinals? Background
I am interested in elementary embeddings from a model of set theory into itself. One way of producing such elementary embeddings is when the model is generated by indiscernibles; this idea is very closely related to the existence of sharps. Jech and Kanamori discuss $0^{\#}$ and $0^\dagger$ in detail but don't tell me much about other sharps. More advanced resources are difficult to understand without a lot of background knowledge.
Hypotheses
Let $\theta$ be an inaccessible cardinal, and suppose that some set $A$ of measurable cardinals below $\theta$ is a stationary subset of $\theta$. For each $\kappa \in A$, let $\mu_\kappa$ be a normal measure on $\kappa$, and let $\mathcal{U} = \{ \langle \kappa, \mu_\kappa \rangle : \kappa \in A \}$. Let $L[\mathcal{U}]_\theta$ denote those elements of $L[\mathcal{U}]$ of rank less than $\theta$.
Question statement
Do there exist large cardinal assumptions which imply the existence of a closed unbounded set of ordinal indiscernibles for $L[\mathcal{U}]_{\theta}$ such that every order-preserving map of these indiscernibles extends to an elementary embedding $j:L[\mathcal{U}]_{\theta} \to L[\mathcal{U}]_{\theta} \, \, ?$
Remarks
The large cardinal assumptions may be on $\theta$, the elements of $A$, or some other large cardinal. The values of $\theta$, $A$, and the $\mu_\kappa$ may be chosen in whatever way you like subject to the hypotheses above -- I just want this to work in some example, not in every example. 
In The Core Model, Dodd mentions double mice, a generalization of $0^\dagger$. Maybe some version of these can be used to answer the question affirmatively, but I know nothing about them.
 A: My reading of this question was different from Andreas', because Norman asked for order preserving maps of the indiscernibles to extend to embeddings 
$j:L[U]_\theta \rightarrow$ $L[U] _\theta$  
i.e. the indiscernibles should be below $\theta$ as the ordinal height of the structures mentioned is $\theta$?
In that case the measurable or Ramsey above $\theta$ only guarantees indiscernibles above $\theta$ and so "missing the target"?    
In any case there are generalisations of "double mice" that you surmise that provide a positive answer. Let $M$ be the "least" in a certain canonical well-ordering of all such iterable structure that have a measurable cardinal $\kappa$ which is, in $M$, the limit of measurables cardinals below $\kappa$.  (Such structures are called "mice".) By Loewenheim Skolem we may assume that $M$ is countable. `Iterability' here means we may form
all iterated ultrapowers of the first structure $M=M_0$ using this top measure repeatedly; call the ultrapower structures $M_\tau$  for all ordinal $\tau$; then all the $M_\tau$ will be wellfounded.  In the $\tau$'th model let $\kappa_\tau$ be the top-most measurable cardinal. Then $\tau ,\mu \rightarrow \kappa_\tau<\kappa_\mu$ and the class of all such $\kappa_\tau$ forms a closed unbounded class of indiscernibles for the model $W$ "left behind" by these iterates.
[Fornally $W = \bigcup _ \tau  H(\kappa_\tau)^{M_\tau}$.]
Then $W$ is the `least' inner model with a proper class of measurables cardinals,
(because as $\tau$ increases the order type of the measurables in the models $M_\tau$ increases; but these measures are left behind in the lower $H(\kappa_\tau)^{M_\tau}$ part
of the model that goes into $W$). The $\kappa_\tau$ are indiscernibles for it, just as the Silver indiscernibles are for $L$.
The large cardinal assumption is then that "This iterable $M$ exists" just as a parallel
to "$0$-sharp exists" is to $L$.
