Once you current shampoo and conditioner run out, try finding a lower-plastic alternative.
Both Faith in Nature and Suma sell robust bottles of shampoo, conditioner and body wash, which can be purchased initially and then topped up at refill stations in many zero-waste shops. Faith in Nature is palm oil free (yay!), whilst Suma uses sustainable palm oil in its conditioner. Sadly Faith in Nature no longer reuses the 5L containers it sends out to the zero-waste shops. Suma sends out 25L containers (i.e. creating less plastic waste than using the 5L containers) but which it also isn't able to reuse. I guess it is best to think of the 'refill haircare option' as lower-waste rather than zero-waste.
If visiting a refill stations doesn't work for you, then a local-to-you zero waste store may sell you the bulk 5 litre+ containers - thanks #OpenHouseGateshead - which will last for ages. You will need a compatible pump, or a funnel, plus a sturdy plastic bottle to decant into - I found an old Ecover cream cleaner bottle to be perfect.
Or you could try 'shampoo bars' (i.e. bars of soap to wash and condition your hair with). I have tried many, many types. Most were awful. Anything that is solid at room temperature is going to need a lot of rinsing to avoid a residue - my hair is just too fragile for that kind of washing.
I do like the Lush shampoo bars, listed here, especially 'Godiva' and 'Honey I Washed My Hair' for £8 each. Don't bother with the tins they sell though (unless you want to travel with them) - you'll need a soap dish for home-use. I like this soap dish from Amazon as it keeps the soap drained, whilst catching the drips, this soap dish by JosephJoseph is excellent (in design and looks) but it is more expensive and it will only fit 'normal' sized (circa 100g rectangular) soap bars in it, as it is slimline in size.
Also worth considering are Beauty Kubes - cubes of compressed shampoo powder, made in Cornwall, that come packaged in a cardboard box. You crush a cube, wet it and form a paste to rub into your hair. It performs just like 'normal' shampoo - yippee! *EDIT* *I found that prolonged (6 month+) use of Beauty Kubes led to a build up of dandruff-like flakes on my scalp, so switching between the refillable Suma shampoo and Beauty Kubes works best for me.* Now down to the particulars... I like the 'shampoo and bodywash' variety in the orange Beauty Kubes box. You get 27 cubes for £8.50, so they would be pricey for those who wash their hair daily, but for an infrequent (twice-a-week) hair-washer (like me), they are affordable. I suggest keeping the cubes in a reusable, plastic, screw-top container so they don't get damp, plus then it is easy to get them out one at-a-time mid-shower.
The zero-plastic conditioner which I use (but not in the summer as the heat makes my hair greasy if I use a conditioner then) is an Ethique soap bar from Holland and Barrett. I use the amazing lime-smelling, green 'Guardian' one, which is labelled for 'dry' hair but it perfectly suits my 'normal' hair. It cost £12.99 for a 60g bar, which sounds expensive but it will last for ages (the website claims it is equivalent to 5 plastic bottles of product - I believe this claim).
I hope you find zero-plastic / low-plastic products to suit your hair, as I have (finally) done!