If you apply for a visa giving the dates and details of both trips (viz. first to Finland and subsequently to Italy), the Finnish authorities are OBLIGED to consider your application for the full duration of the trips applied for (viz. BOTH trips). Therefore, if they grant you a visa at all, it should be for the dates covering both trips.

You need to ensure your application specifies in field 24 that you require a DOUBLE or MULTIPLE entry visa and ensure a proper explanation is provided in field 21. Once you apply for that, Finland is effectively obliged to either approve the visa for the period applied for (subject to the appropriate supporting documentation being provided of course) or else deny the visa outright.

However, if for whatever reason you only apply for and receive a Finnish single entry visa with limited validity, nothing precludes you from applying to Italy for a second Schengen visa prior to using the first one. The only restriction is that you may not hold visas with simultaneous validity. So if for example you get a single entry Finnish visa valid from 01OCT to 10OCT and you intend to travel to Italy from 25OCT to 29OCT, you can apply before the Finland trip for an Italian visa that has validity commencing AFTER the expiry of the Finnish visa as the two visas will not be valid at the same time.

This is not an uncommon issue faced by travelers and in my personal experience, I have always been issued the visa for all planned and documented trips by the first country of application. I have also held multiple Schengen visas, with the validity of the second visa commencing on the day immediately after the expiry of the first one. In fact, I have entered the Schengen area on a 3 day trip commencing the actual day that my visa was expiring (but with another visa commencing the next day already endorsed) and had zero issues once I explained the situation to the immigration officer.