non tech question around 3rd party service provider contract conflicts

Non-techie question, but the organisation I work for (I don't work in IT I work in the risk department), has a contract with a 3rd party provider, who essentially host and manage our network infrastructure. For projects that don't fall into the realms of the base contract, there is subsequently chargeable project work which the same 3rd party do..

However, as a customer of their (the 3rd parties) services, I always wondered, what rights we as an organisation have to consider using an alternative provider to do a piece of work within the Network that would typically be done by our 3rd party who host and manage the network.  This is hypothetical of course, but if you have managed a network before, or managed a contract with a 3rd party who host and manage your network – if they quote you the customer for a piece of work outside your base contract, i.e. chargeable project work – and if you feel the quote they have provided is extortionate for the work involved –  have you any rights to seek alternative quotes for other 3rd party IT specialists, or is there something usually built into your contract with your main 3rd network provider, that all chargeable additional work can only be done by them themselves.

Have you ever had any similar situations? If you do have rights to pursue alternative providers for specific work, are there any other stumbling blocks in pursuing alternative solutions to do the work (i.e. the main contract 3rd party being awkward about this situation?). In an ideal world I assume its better to use the same organisation who host and manage the network, but if there quotes for non mandatory tasks seem unfair, can you go elsewhere?