What’s your best tip for navigating the call centre maze?

This is all helpful and I have used sayto0870, but none of the other suggestions is any good if you have to use the Text Relay support service for the deaf. this is a great service, but time consuming on the most basic of calls due to an operator acting as ‘ears’ for the deaf person and typing what the other person says – sometimes both ways. So all these options still apply and can take so long. Thankfully being a BT customer we can dial 0870 and 0845 numbers for free. Text Relay users get a rebate automatically if they are with BT but not other companies.
I have told companies with national rate numbers that I will not (sometimes cannot) call those numbers as they are not covered by the rebate,. I have been known to be on a call to various companies, trying to sort matters out for hours (literally), as going on hold, being given the wrong number to call – often in correspondence and websites, as well as on the phone. I find myself not wanting to make the call. I use email as much as possible, but I fear some companies such as T-Mobile are no longer offering email correspondence.
I have also found after finding an alternative number on saynoto0870 there is an automated message telling me I will be charged etc etc (effectively the cost of the original number).
I have to say thank you to Text Relay, but sometimes it is hard work making a business call.

Sorry to burst bubbles – but “beating” the IVR will not generally have your call answered any quicker – that is simply not how the queuing system works.
When you press 0 or # or whatever it may transfer you to a general queue – but the very point of the IVR is to triage your call and direct you to the right queue.
You will be transferred to a queue where you will wait for someone assigned to that corresponding skill to answer your call. These will be generalists who will more than likely have to transfer you to the right queue once you have identified yourself and your issue. In all likelihood you will then need to identify yourself and explain the situation again.
I know some IVRs can be confusing and frustrating – but try working with them, rather than going out of your way to prevent your call being correctly routed and then whining about the agent not being able to help you later.

It’s true that there are some IVR systems that work well, but in my experience they’re in something of a minority. Like any package they come in varying types, colours and costs and companies all too often seem to plump for the cheapest. BT – a company whose income would suggest it can afford substantial investment – has one of the worst. It’s not simply that the IVR software is ropey but it utterly fails to triage anything effectively, since it doesn’t ask questions you can answer. OTOH, BT’s live chat system is fairly good and they do ring back when you request it. The Mumbai accent intensity varies, too, and some are quite easily understood.

But all this obscures the simple fact that when a company is supplying a complex product which requires attention they should be obliged to provide accessible and comprehensible contact means.