With COBRA, the 60 day retroactive activation period begins when the employer coverage ends. At most but not all large companies, if you leave during the middle of the month your employer coverage continues to the end of that month. In those cases, the COBRA window begins the first of the following month.
The TCC activation window is 60 days from the date of separation or receipt of TCC notice, whicever comes later. Your retro-activation request must be received by the program administrator within 60 days so account for mail delivery time. Any out-of-pocket expenses already applied to your current plan will transfer to the TCC policy.
ACA compliant policies (both on and off exchange) begin on the first of the month. You should consider 2016 off-exchange plans if you are 100% certain of no premium subsidy. You said you wanted to contribute the max to an HSA. Your contribution limit is prorated based on the number of months you have an HSA eligible health plan.
If you are healthy and have no pre-existing conditions, consider a non-compliant short-term medical plan from the end of the TCC retro-activation grace period to year end. ST plans can start any day of the month and usually have broad provider networks. They are not HSA compatible but the premium savings may offset this. Given your age you may be exempt from the penalty for not having compliant coverage during this period. You are exempt if the cost of the least expensive bronze plan is more than 8.13% MAGI. You claim the penalty exemption at year end on your IRS tax forms. Just be sure you are purchasing a medical plan and not an indemnity plan that only covers accident/specific illness.
Unaffordable Exemption (8.05% MAGI in 2015, 8.13% in 2016): https://www.irs.gov/instructions/i8965/ch01.html