I think that he would have avoided the DLR because of the need to tap in and out but I wouldn't have been surprised if he had set off on foot towards an address he knew
He'd already made at least two Tube journeys (Clapham South to KingsX, and then KingsX change Victoria to Tower Hill) and if he'd jumped the barriers, he'd have been noticed by staff and transport police would have been on to him as a fare dodger - a complication he'd want to avoid - certainly by the second journey when he's had time to think a bit.
He spent a lot of time in quite random parts of south London in the earlier part of the day - possibly just killing time, but this was a pre-meditated attack (he'd sourced the corrosive substance) and might therefore have been planning how to get away (cars caught on ANPR in so much of London, he might have been planning to ditch the car anyhow). With family in London, he may have been very familiar with TfL systems, and have an unregistered PAYG Oyster. Or some other form of contactless payment card. So it would take a while to find a pattern of cards being used at the right time/location
And on Wednesday night, no-one know there was a man-hunt, or what he looked like, or that he had a facial injury. So he might have got up towards a huge transport hub like Stratford, where there are just so many possibilities of where he could have gone.
And presumably the likelihood of his injury causing big problems depended on how much got on him, and whether its a superficial scab or something that's gone much deeper and possibly also compromised his eye.
But wherever he's gone, he needs food. water, somewhere to hide and ways to pay for stuff. And I think that's why police are now addressing some of their message to those who might have assisted him.