Since you were only covered by the HSA-eligible HDHP for part of 2018, your HSA contribution limit will be prorated. The instructions for Form 8889 explain how the prorating works, but essentially you determine how many months you were covered by the HDHP by looking at the first day of each month. In your case, because you were covered by the HDHP only from April 2 - May 31, you can only count one month of eligibility for contribution limit purposes. As you are single and over age 55, the limit for an entire year is $4450, but because you only had one month of eligibility, your 2018 contribution limit will only be one-twelfth of that, or $370.83.
With your employer's contribution of $1300, you are already over the limit of what your contribution limit will be if you do not have any more months this year of HSA eligibility.
You can talk to your old employer about doing an excess contribution withdrawal, and the company might take back the extra, or they might give it to you (depending on company policy). If you get it, you'll have to pay tax on it.
If you are no longer in communication with your old employer, you can talk to the HSA custodian about doing the excess contribution withdrawal to get under the limit. They should send the money to you, and you will need to add this amount to your income next year at tax time.
If you leave the money in the HSA and go over the limit, you will need to pay penalties next year at tax time.
Alternatively, if you continue your HDHP coverage via COBRA (or if you obtain HDHP coverage elsewhere), you will get more months of coverage and your contribution limit for 2018 will increase. If you are covered on December 1, 2018, you have the option of using the last month rule and getting the full $4450 contribution limit (subject to other restrictions; see this answer for more details).
As for your medical expenses, any expenses incurred after the HSA was set up are eligible for reimbursement, meaning that you will be able to empty the HSA on this medical bill (after you remove the excess contribution).
To explicitly answer your question about having multiple HSAs: Yes, you can have multiple HSA accounts, so you could theoretically set up your own HSA account and contribute to that. However, your annual contribution limit applies to all of the accounts, so you'd only be able to do this if you are able to raise your limit with more months of eligibility. Alternatively, your current HSA, which was set up by your old employer, probably has some sort of provision for you making your own cash contributions, so if you are able to keep eligibility and you like your current HSA account, you should be able to make additional contributions yourself.