How to ask for additional time to reconsider an offer after accepting it?

After an interview with a company they offered me a salary matching the one I asked for. However when considering the other benefits I was getting at the current employer (which I wasn't getting at from the other company) I realized my initial request was too low and I asked for more. They agreed to my demand and I accepted this new offer via email and telephone, because I was sure my current employer wasn't going to match it.

When I informed my current employer of the offer I received they came back with a counter offer which is slightly better and is actually a 60% increase from what I am currently getting from them.

Now I would like to ask the other company (not my employer) that I need some time to reconsider their offer even though I already informed them that I accept it because I got a counter offer from my current employer that is very competitive.

Is this a bad idea? And how should I proceed in asking for this additional thinking time while being professional and polite.

Now I would like to ask the other company (not my employer) that I need some time to reconsider their offer even though I already informed them that I accept it because I got a counter offer from my current employer that is very competitive.

Is this a bad idea?

Given that you already accepted the offer, both in email and by phone, going back on your word now would be unprofessional.

You say "I was sure my current employer wasn't going to match it.", and thus accepted the other offer... This was your mistake (calculated risk), as you assumed something, made a choice and it turned out you could have checked with your current employer before accepting an offer.

Now, it is up to you to decide what you want to do. Be careful in being too "greedy", and try to back-and-forth the negotiation, seeking higher payment each time, as you may end up with your current employer not raising and the new company backing up. As they say, "a bird in hand is worth two in the bush".

You say here that you "need time to reconsider" as you got a "competitive counter-offer from your employer", and that you wish to convey that to the prospect company... if you actually say those words to the company, it is very likely it will be taken as "I got a better counter-offer, will you raise yours?", which could end up in them backing off...

If you need to reconsider the pros and cons I suggest you do that immediately, instead of sending an email asking for time to think. If you decide for the counter-offer, then do proceed to tell the company you are backing off... If not, you will have avoided sending an unnecessary email that could have had negative consequences on your job-searching process.