If you want to stay in ArcGIS then lest cost path analysis is your only option but, a massively lacking one. I would highly recommend that you abandon the idea of least cost and reconcile that you need to find a solution outside of ArcGIS. My preference are Kernel approaches, for many more reasons than I can go into on a forum. There is no turn-key GUI software for resistance Kernel models and if you want to go down this road you will have to implement your models in R or Python. You could explore CircuitScape which is a good and well accepted connectivity modeling approach and certainly better than implementing a least cost path analysis (which will not be publishable). There is now a GUI option for CircuitScape with Julia as the processing engine and a large body of literature. If you want to identify explicit dispersal corridors, rather than dispersal potential, then you do not want to use kernel methods and CircuitScape would be the preference.
Please also keep reading the primary literature on connectivity. Using an SDM estimate as your resistance surface will yield some notable issues associated with bias and error. It is also very questionable that you are representing underlying resistance processes that influence how animals move across the landscape and does not track with some of the underlying theory, as lacking as it is, on landscape resistance. The only time this would make sense is where movement and occupancy are the same process eg., animals, like elephants, that continuously move through the landscape. Cushman et al., has a nice paper on this.
Another key consideration, that is rarely mentioned in the literature, is the dispersal capacity of your species. Dispersal limited species are going to have very different movement characteristics, with different resistance parameters, than wide-ranging generalists. This is something that only resistance kernel approaches can directly account for. You can trick CircuitScape a bit by modifying the heuristics and specifying scale dependent parameters but, this is difficult and is somewhat questionable that you captured the intended dispersal characteristics. There is a new implementation of CircuitScape called OmniScape based on a moving window modification of CircuitScape that allows for this but, the methods is still experimental and really is just a poor-mans kernel model. Here is a webmap service from TNC's California chapter showing some state level connectivity analysis using this method and human modification.
Oh, and please rethink MaxEnt for distribution modeling. There are a few papers out now (eg., Renner & Warton 2013) that call into question its nonparametric qualities and prove a mathematical equivalency to point process models and conditional logistic regression. There is also a question of how the probability function is initialized, scaling to an assumption of 0-1. This functionally means that the idea of the resulting probability distribution representing any type of likelihood of occupancy or habitat suitability is nullified. The only supported inference is a threshold probability for presence, not even absence can be supported as the lower tail of the probability distribution represents a there be dragons and not a measure of habitat preference.