Hm-mn.

A softener will remove iron from water, until it's saturated. It sounds like you have a mix of water sources: pre-softener, post-softener in fully regenerated state AND post-softener in un-regenerated state.

You need to bucket test your water, pre-softener. I'm pretty sure you'll see a different result. By the way, if you have iron in your water, you also need to be adding sodium hydrosulfite to your brine, before you regenerate. Iron will permanently foul the zeolite media, and result in greatly reduced life. (Unless, you have a greensand filter, in which case you regenerate with a potassium permanganate solution.)

Regardless, you've got several possible ways to go.

1. Clean up in place

2. Drain & refill, but dose with HEPD and polyquat, so you can gradually remove the iron, without having it all settle out.

3. Drain & refill with post-softener water, stopping to regenerate the softener as needed, to prevent iron 'break-through'.

REGARDLESS of how you go, better filter cartridges (Unicel or Filbur) will make a big difference.

+ The OEM cartridges in Intex / Summer Escapes / etc. filters are not high quality. The Intex manuals recommend replacing them completely every 2 weeks. A BETTER option is to replace them permanently with 2 - 3 Unicel cartridges, and rotating them through a cleaning solution. Properly maintained, the Unicel cartridges will last all season AND will filter better. So even though they cost more per cartridge, they cost less per season! Learn to maintain them, by reading this post, Cleaning-your-Cartridge-Filter-wiithout-Damaging-It!

But, purchasing them is tricky. Amazon does not consistently stock them, and some of the 3rd party sellers are charging high prices and/or high shipping. So, check the Amazon link, but watch out for the prices and shipping costs.

It would be nice if you could just buy them elsewhere, but many of the online pool stores are listing Pleatco or other Asian made low quality filters, as "Unicel C-4607 type" or "fits Unicel C-4607". So shop carefully!

Here are the Amazon links:
Unicel C-4607 Replacement Filter Cartridge (Easy Set Size A or C) @ Amazon
Unicel C-5315 Replacement Filter Cartridge for 15 Square Foot Intex B Filter @ Amazon