Technically, the answer is no, you can't put more money in to that ISA in that tax year once you've transferred it (unless you've transferred it to a Stocks and Shares ISA). FAQs 7 and 8 from http://www.hmrc.gov.uk/isa/faqs.htm cover these cases:
Q. I have transferred current year payments to my cash ISA to a stocks
and shares ISA, can I make any further payments to a cash ISA in this
A. Yes - provided you haven't already used up your annual ISA
investment allowance (£11,520 in the tax year 2013-14).
When you transfer current year payments to your cash ISA to a stocks
and shares ISA it is as if that cash ISA had never existed. Any money
you saved up to the date of transfer will be treated as if you had
invested that money directly in the stocks and shares ISA.
For example, if you had put £2,000 in a cash ISA and then transferred
it to a stocks and shares ISA you would be able to make further
payments totalling £9,520 in that tax year. You could either put all
of the £9,520 in the stocks and shares ISA or you could put up to
£5,760 in a cash ISA (with the same or a different ISA manager) and
the remainder of the £9,520 in the stocks and shares ISA.
Q. How many ISAs can I have?
A. There are limits on the number of ISA accounts you can subscribe to
each tax year. You can only put money into one cash ISA and one stocks
and shares ISA.
But in different years, you could choose to save with different
managers. There are no limits on the number of different ISAs you can
hold over time.
However, in practice, if you transfer mid-year from Provider A to Provider B, Provider B has no way of telling whether you have already put money in to the ISA with Provider A that year or not, so you will be able to put more money in.
I believe that ISA providers do report their subscriptions back to HMRC so they can check for multiple subscriptions (over the limit) in one tax year; but in the past, I have done exactly what you describe and it has never been picked up in any way or caused any problem at all. As long as you stick to total subscriptions within the limit, I'd guess you're unlikely to encounter a problem. (Of course I am not a financial advisor so you should take what I say with the same pinch of salt as you would take any other random advice from the internet).