Can the police track my cell phone if I left the country and went to another one and the SIM card is not supported by the country I went to?
Supported SIM or not, triangulation can still occur from the device pinging off of towers or signals, even if there is no data being moved across those specific mediums. This is an example you can look into, and is something the US has done overseas with drone activity. This is a broad question. Most likely though, it wouldn't be the police.
Another thing to consider is if you do use a network like WiFi, even if it's a new phone: connecting to accounts that can verify your identity (bank, social media, etc.) will also point to you. Very similar to opsec within VPN/Tor usage. If you have more specifics, feel free to ask. Otherwise you're going to get very broad answers.
Like the other answer said, not the police. And why leave it in when it's not supported? Without SIM it is a lot harder to find a phone.
Without a SIM, the handset is not interested in registering on the network. On powerup, the handset will scan the GSM bands to find the BCCHs (broadcast control channels) of the available base stations. On these channels, the base station transmits all necessary data required by the handset to contact the network, in particular the RACH (random access channel) used by the handset to initially contact the base station. If the handset is not interested in registering itself with the network (for which a SIM is required), it remains passive. If you try to make an emergency call, the handset will request this on the most suitable RACH.
yes, they can hunt you down! using the IMEI which is broadcast by you mobile phone with your number when it's looking for a GSM connection ,it's automatic and it's done each time that the gms is connecting to the mobile network even in the rooming, where you are using a sim card of a company X in an other country where this company does not exists, so if you GSM number or the IMEI of your device was flagged, the telecom network can knows that you are at a precise location using triangulation. to avoid this, change the imei or change the gsm AND the sim card. If you change them, they can track you down by monitoring a number that you may contact and trace you back. To avoid this you can use an internet phone number which may take time to be traced which is fare away from some governments jurisdiction and a justification/ sometimes a judge order for that from the country where the VOIP service is represented which is not needed in dictatorships where the companies are forced to deliver what ever they asked for.