Really depends on what lenders you're looking at, as well as your credit profile, income, etc... Be better if you can give more details but, for argument sake, if it's squeaky clean with high scores of 780+, I wouldn't go for anything else for at least 90 days... Otherwise, 6 months minimum. Dont' forget you're also going to be taking hits for the new accounts when they're reported
Good income, but like I said, you're going to be taking hits for the new accounts, so I'd wait and see where that all settles. Personally, I'd wait until I was 700+ again before thinking about any more new accounts, how ever long that takes. You can always play the AZE0 game and go for CLIs on your existing accounts in the meantime. In this game, when people can't put the brakes on themselves, the system will put them on with toy limit approvals or just outright denials.
You should cool it for 12 months!
Your score is going to be under 700, therefore any new cards that you get would have toy limits or denial.
With your score, you don't want to be labeled as "Credit Seeker" and remain with "Poor Credit" for next couple of years.
The best thing you can do for your credit is to garden for a year and let positive history report. Once your score is mid 700, then go after a better card that you can build on.
Good luck.
Congratulations on paying down your debt and generally acting right with your new usage, nicely done!
AZE0 is all zero except one... only let one card (a revolver) report a small balance. Lots of threads here on the subject.
As far as your Amex dilemma I can only speculate that it's a combination of your score as well as past high usage/keeping high balances on your other accounts. They will continually SP your credit (which is usually ip to 60 day old data)... I'd just keep up what you've been doing and hopefully they'll eventually loosen the reins.
Welcome to My FICO forums, @IWO. 
Some of the discussion above was more focused on how much credit-seeking someone can do before it impacts approvals, which was one portion of your question. But you were also asking about FICO impact on inquiries. Unfortunately, there's no hard-and-fast rule on how much each inquiry will affect a profile since there are many variables. FICO has multiple "score cards" for different types of credit profiles and then there are further differentiating factors such as AAoA (Average Age of Accounts), AoOA (Age of Oldest Account), number and types of accounts (revolving and installment such as auto or mortgage), and derogatory information to include collections, bankruptcy, or late payments. A thin (younger or less developed profile) would generally have more of an impact from a new account or inquiry than a thicker, developed profile. In either case, the FICO penalty increases with more inquiries, so less is better.
Yes, the FICO penalty is temporary and FICO rises slowly as new accounts/inquiries age. Even though they stay on your credit report for 24 months, they no longer are "scorable" (affect FICO) after 12 months. You can read more or ask questions about the FICO effect in the >"Understanding FICO Scoring" sub-forum<.
Don't confuse the impact on FICO with the probability of new card approval! FICO is only one part of the approval equation, and sometimes you may get a denial even with high FICO!!
As a general rule-of-thumb, I recommend most consumers strive for a benchmark of no more than about one new account/inquiry every six months. This will keep you within the majority of lender's tolerance for new accounts (such as Chase 5/24 or BofA 3/12) without having to track each one. The higher velocity above that, the greater the chance for denials or lower "toy" limits as mentioned above. That holds true for all credit profiles to some degree, but a mature credit profile is more resilient and may get away with a little more aggressive credit-seeking. We see a lot of newer members who are trying diligently to rapidly increase their credit, and who push too hard. Patience is the key. We like to remind forum members that credit is a marathon, not a sprint.
Toy limits aren't strictly defined, but IMO are typically $1000 or less. But what happens, and I've seen this even with a mature profile, is that excessive credit seeking will result in lower limits than could otherwise be achieved with the same profile. When I push too hard, I've watched my starting limits decrease and sometimes have even gotten denials with 800+ FICO, high income, and mature credit profile just because lenders were uncomfortable with my pace of new accounts. So patience means that you'll get more "bang for the buck" on each hard inquiry+new account with better SLs on each one and less denials.
